2020
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# All geographical distances are optimal

Toutes les distances géographiques sont optimales
Todas las distancias geográficas son óptimas
Alain L’Hostis

## Abstracts

Triangular inequality is one of the four mathematical properties of distance. Its respect derives from the optimal nature of the measurement of distance. This demonstration (L'Hostis 2016, 2017) reveals key aspects of distances and geographical spaces. We develop this argument by investigating the idea of the optimality of distance through a mathematical and geometric discussion, and by dealing with empirical approaches to applied geography.
The first part of the paper explores the consequences of considering that the mathematical property of triangle inequality is always respected. In fact, no violations of triangle inequality are observed in geographical spaces. The study of the optimality of distances in empirical approaches confirms the key role of the property of triangle inequality. The general principle of least-effort applies for most movements and spacings. In addition, however, trajectories with multiple detours, like those of shoppers, runners and nomads, are optimal from a certain point of view. This is also the case for excess travel, i.e. a situation of disjunction between an optimum as perceived by a person in movement and an optimum as perceived by an external observer. Any movement, any spacing within and between cities and in geographical space in general, exhibits a form of optimality, and all geographical distances are optimal.

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## Index terms

### Mots-clés :

distance, transport, théorie spatiale

### Keywords :

distance, transport, spatial theory

### Palabras claves :

distancia, transporte, teoría espacial
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## Full text

The author thanks the anonymous referees that helped to improve the article through their remarks and reflections. The paper also benefited from the reading by a mathematician, Benoît Kloeckner.

# Introduction

1Geographical distances are measurements of the separation between geographical objects. Two kinds of geographical distances exist. It can be a geometrical computation – such as Euclidean distance – that generally takes the form of a formula (analytic approach), or it can be a measurement along an itinerary. This second type of measurement can be derived from the output of a graph algorithm (in general without analytic formula) or come from an empirical measurement.

2In the study of geographical distances, discussion of the linkages between mathematical properties and empirical geographical or economic observations has proven fruitful. In particular L’Hostis (2014) has demonstrated that triangle inequality, which is one of the four mathematical properties of distances, reveals several key features of distances.

3Distance is defined in mathematics as a function which, for any two given locations a and b in a given space, obeys the following properties, also called metric properties:

P1 (positivity) d(a, b) 0

P2 (distinguishability) d(a, b) = 0 if and only if a = b

P3 (symmetry) d(a, b) = d(b, a)

P4 (triangle inequality) d(a, c) d(a, b) + d(b, c)

4The first three mathematical properties of distance have no major implications for spatial analysis (L’Hostis, 1997, p. 114). Even symmetry, a property almost absent from empirical spaces, brings nothing radically new in the way space is considered or modelled. The fourth, triangle inequality, states that for a given distance between A and C, no alternative smaller measurement along a different path through B can be found, whatever B. There is no shorter, faster or cheaper route, according to the criteria considered in the distance function. The triangle inequality is observed any time the distances measured between pairs of places, in a context of measurement additivity, returns the smallest possible value. In other words, the triangle inequality is observed as a consequence of the optimality of distances. The term optimality here refers to the idea that the effect of measuring distances is to optimise a criterion, for example, time, kilometres or cost. For a distance measurement between any two given places to be optimal, it has to produce the shortest possible measurement. In this sense, the hypothesis of this paper is to state that distance is always a minimum measurement, and to start exploring the geographical meaning of this mathematical statement. Geographical distances are always produced by the rational choice of individuals or groups. In the scope of this paper, we will consider that these choices and the associated measurement reflect a rational intent (Wolpert, 1964, p. 558).

5L’Hostis’s analysis of errors in the interpretation of violations of triangle inequality (L’Hostis, 2016) leads to the observation that geographical distances are optimal and that detours and breaks contribute to this optimality (2017). As distances can serve to define geographical spaces, understood as bidimensional geographical units, this discussion has broader implications, beyond transport geography alone. In this paper, we develop these ideas by discussing the issue of the optimality of distances in the context of applied geography. More broadly, our discussion on the optimality of distances helps to highlight the role of the concept of distance in relation to the movement and the spatial distribution of geographical objects.

6It may come as a surprise to observe that, in the academic disciplines that focus on spatiality, the scientific literature on distance itself is nothing like as extensive as the use of the concept itself (L’Hostis, 2014). The situation created by the 2020 pandemic which made concrete for everyone the need for distance between people even reinforces this surprise. Indeed, the list of contributions concerned explicitly with distances (Deutsch, Isard, 1961; Hall, 1969; Gatrell, 1983; Huriot, Smith, Thisse, 1989; Brunet, 2009) is relatively short. Furthermore, for many scholars in the fields of spatial analysis relating to economics, geography and planning, distance is not considered to be a central concept. Several factors contribute to the marginalisation of distance.

7Firstly, in the euclidianist conception denounced by Lévy (2008, p. 83), distance is associated with the Euclidean straight line. Following this rationale, challenging the straight line as a general model for routing between geographical locations casts the doubt on the capacity of distances in general to express the relative position of locations in space.

8The second argument states that distance is too multifaceted for ambiguity to be avoided in scientific discourse. Because of its multiple forms, it can be used in ways that are too different from each other. Distance is the length of a route that can be measured with different units, but distance is also simply the idea of separation between locations, or the length of a linear infrastructure. In order to illustrate the multifaceted nature of distance, we can examine the role of distance in postmodern scientific literature. The ideas of Hall (1969), who highlighted differences of perception and the relativity of physical distances between cultures, fuelled and inspired the postmodernists (Steadman, 1996). The analysis of the influence of distances on human behaviour has supported relativism in urban studies and in urbanism in particular, in a general climate of opposition to functionalist modernism in which Jacobs was a key player (Jacobs, 1961). At the same time, a consensus places the idea of distance at the heart of geography. The first law of geography enunciated by Tobler that “everything is related to everything else, but near things are more related than distant things” (Tobler, 1970, p. 236) demonstrates this role. Yet, this idea that everything is related to everything else creates a link between geographical objects and posits a comprehensive vision that Sui sees as running counter to postmodern thinking, which separates, distinguishes and relativises, and rejects the idea that individuals and social groups are effectively or potentially linked to each other (Sui, 2004, p. 271). These two streams of thought, though both inspired by the idea of distance, generate opposing views under the current paradigms. From this perspective, distance, as a concept used in multiple, diverse and even conflicting kinds of analysis, needs to be put aside.

9A third argument, found in the geographical literature (Dumolard, 2011), sees distance as an abstract notion independent of behaviour and hence lacking relevance to human geography or to urbanism. In this view, distance is a meaningless concept in geography.

10In opposition to these three positions, we argue that the notion of the optimality of distances can help to clarify the concept of distance and to assert its role in spatial disciplines. While many scholars do not consider that optimality is part of distance, or that distance can exist out of optimality, as shown in the discussion about the violation of triangle inequality (L’Hostis, 2016), our hypothesis is that optimality is an inherent property of distances. Testing this hypothesis and drawing perspectives is the scope of this paper. This test will be conducted by a mathematical and geometrical discussion and by studying empirical approaches.

11In order to test this hypothesis, we start by exploring the consequences of respecting triangle inequality for the understanding of distances and space. We will then focus on empirical approaches relating to the optimality of distances.

# Triangle inequality respected: coherence of distance and space

12It has been demonstrated that non-respect of triangle inequality poses major difficulties for cartography (L’Hostis, 1997, p. 116). This is because it breaks the continuity of the geographical surface, which translates into topological breaks: this is the problem of space inversion formalised by Bunge (1962, p. 171) and Tobler (1961, p. 106), which is very common in geographical spaces (Gatrell, 1983, p. 46). Characterising these transformations can be made by referring to homeomorphism, and the mathematical investigation about which properties, including metric properties, are preserved during the transformation (Tobler, 1961, p. 10). Space inversion refers to the phenomenon whereby a traveller takes benefits from starting his journey by setting off in the direction opposite to his end destination, in order to access a rapid transport system like an airport, a high-speed rail station, or a motorway. In this article, we will build on work on triangle inequality in the domains of economics and geography.

13The economist Tony Smith (Smith, 1989, p. 5) credits the mathematician Fréchet (1906, 1918), who was the first to formalise distance and its four properties, and demonstrating that triangle inequality is the fundamental property of metrics. For him, the function indicating a measurement of the separation between two points is a spread (écart) and becomes a distance only if it respects triangle inequality (Fréchet, 1918, p. 55).

14In the domain of spatial economics, Smith showed that any measurement based on minimum paths obeys triangle inequality (Smith, 1989, p. 15), which means that constructing spaces that violate triangle inequality entails creating links between locations that are not minimum paths. As an example of distance violating triangle inequality, Smith cited the discrimination distance between objects seen by a radar (Smith, 1989, p. 7). This type of distance deals with dissimilarities between objects and not with physical separation. In the same spirit, Gatrell also introduced a measurement that violates triangle inequality, with a non-spatial index of dissimilarity (Gatrell, 1983, p. 38), and Felsenstein discussed the possibility of non-metric spaces with respect to the separation between living species (Felsenstein, 1986). In all these examples, the notion of a path does not have the same meaning as commonly accepted in geography, taking us away from distance as understood in transport and geography. This is a key observation, because the layout of a transport network may include direct routes that are close to the straight line but nevertheless sub-optimal. In this case, the layout of the network creates problems for a person planning a route, since there is a strong psychological preference for the direct route, though it may have been superseded by a fast transport system (Mathis, Polombo, L’Hostis, 1993; L’Hostis, 1996, 2009). Nevertheless, such a situation does not lead to violations of triangle inequality (L’Hostis, 2014).

15In the majority of works dealing with international flows, triangle inequality is considered to be obeyed in cross-border arbitrations (Lamure, 1998; Eaton, Kortum, 2002, p. 1745). The only exception found in the literature, according to Behrens et al., is that triangle inequality may be violated in the case of non transport-related transaction costs (2007). In this situation, it would be possible for a matrix of general costs of interactions between locations, combining transport and non transport-related costs, to generate a metric space which does not respect triangle inequality. It is true that, in international flows, transport costs are not the only factors that contribute to distances. So-called transaction costs cover a variety of factors associated with geographical and economic borders, including political, historical, or cultural aspects: a language difference, for example, can reduce economic exchanges between two countries by half (Rietveld, Vickermann, 2004, p. 241). Customs duties in particular can account for a significant proportion of cost-distances between countries.

Figure 1: Triangle inequality violation for custom duties between countries, a transitional situation

Source: A L’Hostis 2020

16Nevertheless, these costs have no connection with the distances in kilometres that goods travel. They arise from factors linked to national or supra-national economic policy choices. Under these circumstances, it is in theory possible to imagine a violation of triangle inequality relating to these transaction costs. In Figure 1, we see three countries A, B as producers and C as an importer of goods. Customs duties are high for goods entering country C directly from country A, but much lower from country A to country B and from country B to country C. In this case it is possible that the sum of customs duties AB and BC remains lower than the direct cost from A to C. The hypothesis of cross-border arbitration implies that this situation will not last, because the intermediate country B will become a privileged entry point in C for goods produced in A. This means that the economic flow will seek to minimise transaction costs by establishing a path with reduced costs. It would be an appropriate economic policy for country C to reduce the custom duties of direct import from A or to increase the fares from B in order to charge the goods originating from A through their direct or indirect path. In mathematical terms, this process means that the system will obey the property of triangle inequality, or will correct violations in triangle inequality, which cannot persist in the international system. In addition, the assertion that this situation represents a violation of triangle inequality can be challenged: the distance between A and C will follow the route through B, and there exists no other shorter or cheaper measurement. In that sense route AC is not a measurement of distance, but simply a measurement on a path.

17Huriot, Smith, and Thisse proposed to define distance in spatial economy as the minimum-cost distances measured along minimum-cost path (Huriot et al., 1989). In this case optimality is directly derived from the way distance is computed. Stating that such distances are always optimum is a tautology because optimality is imposed in the distance measurement. Nevertheless, we notice that the proposal by the three spatial economists is coherent with our general assertion. And we observe that they felt the necessity to introduce the notion of optimality in their formulation of distances, which is also a mark of the importance of the notion in the definition of geographical distance.

18With this first part of our analysis, we find in spatial economics strong grounds to justify obedience to triangle inequality, and hence to support our statement about the optimality of distances. Let us now move from the domain of economics to the more applied topic of transport and geographical networks. According to Ahmed and Miller, in the renewed time-geography framework (Miller, 2005; Shaw, Yu, 2009) as applied to the domain of transport, the only metric property whose violation is ‘likely’ is triangle inequality (Ahmed, Miller, 2007). Nevertheless, their own study of time-space in Salt Lake City conversely shows that triangle inequality is not violated. Ahmed and Miller consider triangle inequality to be ‘likely’ but they do not provide evidence of this likelihood. In addition, their work mentions Tobler’s conjecture according to which using only minimum paths permits avoiding any violation of triangle inequality (Tobler, 1997). This view carries an unsolved contradiction, and it does not convince that such violations exist in geography.

19More generally, the contributions on graphs, whether geographical or not, provide a very relevant analysis for understanding the nature of geographical networks (Mathis, 2007). Schilling, Rosing and ReVelle observe that the two main characteristics that allow us to distinguish what they call “natural networks” from random networks (2000) are symmetry and respect for triangle inequality: for the authors, “natural networks” are symmetrical and respect triangle inequality. In their paper on the development of location algorithms for the resolution of the p-median problem where a location for p facilities must be found in a given network, they show that imposing triangle inequality is the factor that brings the fastest convergence towards a solution. When solving the issue of the best location for a set of facilities in a random network, the imposition of triangle inequality on the network is the factor that has the strongest impact on the efficiency of the location algorithm. This argument supports the idea that geographical networks, and hence geographical distance, possesses special characteristics and properties. This is yet another argument in support of the importance of the property of triangle inequality for geographical space.

20In the domain of mental maps, a flourishing field in the era of plastic spaces (Forer, 1978), a major aspect of the work consists in explaining the differences between mental, perceived or cognitive space, and geographical space (Tobler, 1976; Gatrell, 1983, p. 130). Psychological analysis of the representation of space and distances reveals that violations of triangle inequality can exist for some subjects (Cadwallader, 1979; Baird, Wagner, Noma, 1982, p. 205). Some of these violations can be attributed to individual deficiencies in spatial knowledge: our mobility routines leave large parts of space unknown (Gatrell, 1983, p. 132). Nevertheless, these gaps in knowledge are not significant and the level of error is low (Moar, Bower, 1983), which leads Baird, Wagner and Noma to argue that such deviations could be understood as measurement errors rather than evidence that cognitive spaces are not metric (Baird et al., 1982, p. 205; Golledge, 1997, p. 238). According to this analysis, there is no strong evidence to support violations of triangle inequality in the domain of mental maps. More generally, as in all empirical approaches, the issue of measurement errors has to be dealt with distances measurement. The choice is then left to the analyst to determine whether the error hides factors not taken into account, as in a systematic bias, or if the error is generated by the approximation of the measurement tool. In this case, and according to the consensus expressed in the literature on mental maps, we adopt the point of view of the second choice: deviations of perceptual or cognitive space to the triangle inequality can be considered as measurement errors.

21Tobler introduced a number of methods for building mathematical spaces from empirical data obtained through measurements on the transport system (Tobler, 1997). Using the length of routes between cities in mountainous western Colorado, he built distances approximated by several methods derived from bidimensional regression. For Tobler, if the measurements of separation between adjacent cities are minimum, then the distance produced will obey triangle inequality (Tobler, 1997). In his thesis, Tobler equates violations of triangle inequality with spatial inversions by asserting that “a place located two hours away cannot be closer than a place situated one hour away” (Tobler, 1961, p. 120). From a geographical perspective, for Tobler, violations of triangle inequality are spatial aberrations. We can add that this principle also applies in a cost space.

22In this second part of our analysis, we observed that geographical networks are not random networks and possess key properties, including the obedience to triangle inequality.

• 1 We consider that measurements of distances on the sphere, through geodesic curves or great circle, (...)

23In order to complete this investigation into triangle inequality, we will now study the property of transitivity in geographical and social networks. Social space is primarily defined by the relations between individuals and groups (Bourdieu, 1989, p. 16) and does not need to consider their position in geographic space The mathematical models of network-distance maintain transitivity, as in Bae and Chwa (2005), who define the general mathematical properties of network-distances superimposed on Euclidean distance measured on the sphere,1 which respect triangle inequality. Transitivity, which is maintained by triangle inequality, is a property of locations in geographical space that is not always observed in social space: although individual A knows individuals B and C well, it is possible that the latter have no direct link to each other. In social network space, A is close to B and to C, but B and C are very far away from each other. In this case, the distance between B and C is greater than the sum of distances AB and AC, causing a direct violation of triangle inequality. However, it has to be said that this situation depends on how distance is measured.

24There are two types of measurement: we can measure the frequency or the intensity of direct contacts, or we can count the number of intermediate people between two individuals. In the case where distance represents the inverse of the volume of contacts, with no contact corresponding to an infinite measurement and multiple contacts being equivalent to a short distance, many violations of triangle inequality are possible. This measurement is an écart according to Fréchet (Fréchet, 1918, p. 55). As this measurement is consistent with the idea of proximity between individuals, we can verify that social spaces violate triangle inequality, which we interpret as attributable to their being only partially determined by spatial arrangement. In terms of spatial properties, social space cannot be considered equivalent to geographical or transport space. A distance can then be constructed from these measurements by minimising path in the social network. This leads to measures like counting the number of intermediaries, as in small world approaches (Watts, 2003), and in this case transitivity is obeyed.

25As has been highlighted concerning non-spatial dissimilarity measurement, social non-spatial networks may exhibit violations of the triangle inequality. Triangle inequality violations can be found in non-spatial sets, but cannot be observed in spatial networks. This argument reinforces our statement that geographical distances always obey triangle inequality.

26It must be recognised that erroneous conceptions are easier to find than correct ones. In this second group we include, for example, Lévy who, in his critique of the Euclidean approach in cartography, illustrates his argument with the path through B that is faster than the direct route from A to C (Lévy, 2008, p. 83). In so doing, he makes a correct analysis of the often wrongly interpreted (L’Hostis, 2014, 2016) property of triangle inequality. Another illustration comes from Miller and Wentz in their paper on the formalisation of measurements in spatial analysis: they present the issue of the measurement of length by discussing the properties of distances, but at no time do they cite geographical spaces where triangle inequality is violated (Miller, Wentz, 2003).

27In this discussion on the property of triangle inequality, we observe the emergence of the notion of the optimum. We will now shift our examination to this notion.

# Distance and the optimum: lessons from empirical approaches

28In this discussion we find two opposing ideas about the role that distances might play in the reflection on geographical space. On one side, distance is considered as a neutral quantity devoid of predefined properties, as in the example given by Haggett (2001, p. 248), reproduced in figure 2, in which a non-optimal measurement of distance is introduced. The measurement between q and s is suboptimal since a shorter route through r exists; this 6 hour measurement cannot be considered to be a measurement of distance. Haggett most probably inserted here the duration of the shortest path in kilometres, which introduces a suboptimal measurement and an inconsistency in space and distances, expressed by the violation of triangle inequality. According to this analysis, the duration is a parameter of the path which is not used in the computation of the optimum path in kilometres, and hence in the optimisation process of the distance. The duration of the fastest route would have been relevant in this diagram, and would have proposed a geographically coherent representation.

Figure 2. Four cities with non-optimal separation measures, violating triangle inequality

Source: (Haggett, 2001, p. 248) underlined by the author

29On the other side, some see space, and hence also distance, as a reality that cannot be separated from the way individuals perceive it, to the extent that they refuse to include it as a fundamental dimension in spatial disciplines (Chivallon, 2008). This discussion is influenced by the debate in sociology between those who do not consider geographical space as a central factor of explanation of social facts (Chamboredon, Lemaire, 1970), and those who propose to rebase their discipline on mobility and hence on space and distance (Urry, 2008, 2012).

30Here, we propound a third idea, which is that distance in geography cannot be considered a neutral quantity devoid of any mathematical properties. Distances embody a set of geographical properties – analogous to temperature or altitude – obey constraints and laws, and reflect a spatial organisation determined by geomorphological, historical, climatic, perceptual or cognitive dimensions. When considering the mathematical definition of distance, we must accept that triangle inequality is respected, due to the fact that distance is the minimum of the measurements of separation between two locations (L’Hostis, 1997, 2014). From the mathematical perspective, minimality is a fundamental property of distances.

31The optimality of distance is justified on robust theoretical grounds, but we will now discuss optimality in empirical approaches. We will consider observations in the social domain, with a focus on movements in space. Mobility analysis, transport modelling and, more generally, empirical approaches, shed light on the role of optimality in distances. From this analysis we will then discuss the way optimality is incorporated into spatial concepts.

Figure 3. The desire line revealing tensions in the formation of distances as a compromise between least-effort and risk associated with road safety

Source: picture L’Hostis 2013

33Sociologists have introduced the term desire lines to describe the visible paths generated by walkers in landscapes (Simmel, 1997, p. 171). These desire lines, as the one visible in Figure 3, reveal the tensions in the formation of path and hence in the formation of distances, which are often the result of a compromise between least-effort and considerations of road safety. It therefore seems legitimate to focus on the influence of the least-effort argument in pedestrian routes in general: to what degree is the least-effort trajectory favoured in general situations? In fact, desire lines are revealed under particular conditions, namely the existence of a surface which, as in the example of grass covered soil, is able to record and display pedestrian flows.

Figure 4. Optimisation in pedestrians routes in a shared space in Ashford (UK); in blue, straight line pedestrian crossing, in red pedestrian trajectories including detour to formalised crossing

Source: Moody and Melia 2013

34Studies based on video analysis, such as Moody’s and Melia’s work on shared space in Ashford in Great-Britain (Moody, Melia, 2013), show that this practice of pedestrian route optimisation is far from an isolated case. In this space, shown in Figure 4, almost half of the pedestrians routes favour a more rectilinear trajectory – a desire line – which is more dangerous than the detour through formal road crossings, despite the fact that the shared space design principle was supposed to make them obsolete. The optimum and the relation to the straight line influence these route choices: the desire line is the manifestation of a deep-rooted facet of human behaviour.

35Nevertheless, outside all these situations, it can be objected that not every trajectory in geographical space seeks to minimise effort. The list of these cases is long: a running course, a trekking route, a tourist trip, the random walk of ants (Berg, 1993), the stroll of the shopper or the wandering of the nomad. These kinds of movements rarely minimise the time spent or the number of kilometres travelled, or, more generally, the parameters of the effort needed for the movement. But all of them optimise a different parameter: pleasure in strolling for the flâneur (Tester, 1994) or the tourist, physical achievement for the runner, maximisation of the number of opportunities to find a resource for the nomad or the shopper. All these cases seem to depart from the least-effort principle, if we consider for instance the extra effort required by the shopper in his/her meandrous itinerary as compared to the shortest path between origin and destination, but they do not invalidate the hypothesis of optimisation. All these trajectories are optimal. Optimisation can occur along the path, as in the trajectory of the flâneur, or can focus on the extremities of the path, as in the case of a race. Whether or not they should be considered as distances is open to discussion: since shorter routes exist that meet a different set of criteria, it is possible to find measurements that would be shorter. In this sense, these measurements do not exactly represent the amount of time or space separating an origin and a destination, but instead only describe a complex optimum path in space. While undoubtely describing actual human practice of space, these complex optimum path situations would therefore seem to lie outside our scope for the study of geographical distances.

36In this first part of the empirical analysis, we observe that least-effort, an optimisation process, plays a key role in geographical distances. The observation of pedestrian movement and other types of movements in space confirms this idea. We will now consider conceptual approaches of mobility.

37As a further illustration of the process of optimisation in mobility, we consider the notion of time-budgets, the fact that individuals devote a given and relatively stable amount of time to mobility in a daily basis (Zahavi, 1976). The debate on the constancy of time-budgets and the increase in travel distances shows that individuals optimise their time by maintaining the personal time-budget they dedicate to mobility by adjusting the distances travelled (Banister, 2011). We can see this phenomenon as a process of distance optimisation in which individuals engage on a day-to-day basis.

38In traffic engineering and network economics, many models are developed by analogy with physics, especially fluid dynamics: traffic flows exhibit properties and behaviour that are suitably described by fluid dynamics (Leurent, Chandakas, Poulhès, 2011; Ma, Lebacque, 2012; Costeseque, Lebacque, 2013). In this domain too, optimality is an essential principle, explicitly related to Fermat’s principle, according to which light propagates locally along a route that minimises trip duration. In this domain, the reference to the optimisation of distances is explicit.

39Returning to mobility analysis, the theme of excess travel emerged from doubts concerning the widely accepted economic definition of transport as derived demand, i.e. demand that has no intrinsic justification and only exists for the sake of the activity that it permits at the destination (King, Mast, 1987; Salomon, Mokhtarian, 1998; Mokhtarian, Salomon, 2001; Kanaroglou, Higgins, Chowdhury, 2015). Excess travel refers to trips that are longer than the optimal routes identified by computation (King, Mast, 1987), and to trips made for the pure pleasure of movement (Mokhtarian, Salomon, 2001). However, this literature highlights the disparity between the itineraries chosen by car drivers and those evaluated as optimal through external measurement. The thematic of excess travel also tends to oppose the microscopic optimum of the individual route choice, to the macroscopic urban optimum in the context of congestion. This idea does not undermine the notion of the optimum for individuals, who can make decisions in full or partial cognizance of objective travel conditions. The focus is more on the disparity between an optimum perceived by individuals and an optimum measured by external observers. The issue of excess travel does not undermine the thesis of distance as an optimum.

40Many authors have argued that people need separation between home and the workplace (Lynch, 1981, p. 194; Lynch, Rodwin, 1958). In surveys, the vast majority of people interviewed do not want to do away with the time-budget that they dedicate to transport, explaining their choice by a need to separate home and work. This desire to separate functions and activities reflects a wish to maintain the distinction between private life and work life (Belton, 2009; Pradel, Chevallier, 2010). This hypothesis of separation again does not challenge the idea of optimality; indeed the need for distance corresponds to a form of spatial optimization. In this view urban geographical distances not only consider movement but also the possible change in the location of places that generate urban movements, e.g. homes, workplaces or shops. In this case the spatial optimization process engages both the routing between places and the possible relocation of urban functions.

41Optimum distances should not, however, be confused with desired distances. All the specialists in mobility acknowledge that most people, especially in big conurbations, complain about travel condition and travel time (Milakis, Cervero, van Wee, Maat, 2015). They would like their travel time budget to be smaller, and do not perceive it as optimum. Nevertheless, given their constraints, particularly the economic constraints on their choice of residential location, their mobility is optimised. Their travel-time budget is as limited as they can afford and the hypothesis of geographical distances as optimum still holds. The optimisation process here refers to the complex set of choices of home location, of transport mode, of itinerary and of other movement parameters like time of departure.

42The principle of least-effort is now used in the modelling of pedestrian mobility (Guy et al., 2010), or in the analysis of street networks at the scale of conurbations (Masucci, Smith, Crooks, Batty, 2009). We have discussed, in relation to desire lines, the consequences of neglecting the least-effort principle in urban design: users may develop their own networks and ignore the itineraries designed for them by planners. In the study of mobility, effort and its minimisation come into play at a basic level for the understanding of desire lines (Kellerman, 2012, p. 27), but there has been little significant analysis of the topic, and, alternatively, other factors tend to be emphasised: individual factors like wealth, age, gender, or trip related factors like purpose or transport mode choice are often considered as the most significant to describe and explain mobility. In geography, many authors recognise that mobility choices reflect a human desire to minimise distances (Gatrell, 1983; Golledge, 1997; Wolpert, 1964). Individuals may not be omniscient and fully rational beings, but their mobility choices reflect a rational intent (Wolpert 1964, 558).

43Empirical analysis shows how important the optimality of distances is for movement and spatial distribution. The idea that all distances are optimal has implications for the understanding of and action within geographical spaces. This suggests that the quest for explanations of any apparently sub-optimal movement should preferentially be conducted within a framework where a form of optimality can be developed, rather than simply assuming the sub-optimality hypothesis. In terms of action, sub-optimality in geographical movement, for example in excess commuting, should not necessarily be seen as an anomaly to be corrected but rather – on the assumption of a rationale – as a coherent choice based on motives that need to be studied. In this sense the sub-optimality argument tends to hide the underlying reasons of the observed phenomenon; the assertion of the optimality of geographical distances help to formulate an alternative explanation based on an opposition between an objective and a perceive optimum.

# Conclusion

44Beginning with the observation of the difference that exists between the broad use of distance in the literature of spatial disciplines and the limited scholarly attention to the concept, we argued that a reflection on optimality could clarify its role and function. We consider geographical distances defined as measurements of the separation between geographical objects. Two kinds of geographical distances exist: geometrical computation like Euclidean distance, or measurement along an itinerary. The first step is to explore the consequences of considering, in contradiction with a significant proportion of the existing literature, that the mathematical property of triangle inequality is always obeyed. It is indeed the case that violations of triangle inequality are not observed in geographical spaces. This reflection led us to focus on optimality, which is the property that is tested by the observation of triangle inequality. Optimality of distance expresses the idea that, in the measurement of separation, one or a combination of criterion is optimised, usually but not exclusively, time, kilometres, or cost. Transport economists express this principle with the notion of generalised cost.

45The study of optimality of distances in empirical approaches confirms its role as a key property. The general principle of least-effort applies to most movements and spacings. Nonetheless, trajectories with multiple detours, like those of shoppers, runners or nomads, are also optimal from a particular point of view. This is also true of excess travel, defined as the disparity between the optimum as perceived by the traveller and that perceived by an external observer. Beyond the cases found in the literature, which – though instructive – fall well short of covering all the phenomena, our reasoning is supported by the fact that optimality is a key property of distances. Any movement, any spacing within cities, or in geographical space in general, exhibits some kind of optimality.

46Counterintuitive though it may seem, the optimality of distances is supported by the existence of detours and breaks in trajectories and movements. Work remains to be done on whether and to what extent spatial theories, in geography and in economics, take into account the idea of the optimality of distances. In terms of application, the key recognition of the optimality of distance has numerous direct implications for transport geography, urbanism and spatial planning. Our analysis indicates that in the quest for improving urban form, transport networks and individual behaviours, the identification of sub-optimal situations to be improved, as in excess commuting, should be carefully tested against the principle of the optimality of distances. This suggests avenues for potential further work on distances.

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## Notes

1 We consider that measurements of distances on the sphere, through geodesic curves or great circle, exhibit many differences with measurements on a flat surface, but do not differ from geometry on the plane with respect to triangle inequality. Hence, the conclusions remain valid in both geometrical contexts.

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## List of illustrations

Title Figure 1: Triangle inequality violation for custom duties between countries, a transitional situation
Credits Source: A L’Hostis 2020
URL http://journals.openedition.org/cybergeo/docannexe/image/35048/img-1.png
File image/png, 59k
Title Figure 2. Four cities with non-optimal separation measures, violating triangle inequality
Credits Source: (Haggett, 2001, p. 248) underlined by the author
URL http://journals.openedition.org/cybergeo/docannexe/image/35048/img-2.png
File image/png, 138k
Title Figure 3. The desire line revealing tensions in the formation of distances as a compromise between least-effort and risk associated with road safety
Credits Source: picture L’Hostis 2013
URL http://journals.openedition.org/cybergeo/docannexe/image/35048/img-3.png
File image/png, 1.5M
Title Figure 4. Optimisation in pedestrians routes in a shared space in Ashford (UK); in blue, straight line pedestrian crossing, in red pedestrian trajectories including detour to formalised crossing
Credits Source: Moody and Melia 2013
URL http://journals.openedition.org/cybergeo/docannexe/image/35048/img-4.png
File image/png, 1.6M
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## References

### Electronic reference

Alain L’Hostis, « All geographical distances are optimal », Cybergeo : European Journal of Geography [Online], Systems, Modelling, Geostatistics, document 947, Online since 24 June 2020, connection on 13 August 2020. URL : http://journals.openedition.org/cybergeo/35048 ; DOI : https://doi.org/10.4000/cybergeo.35048

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### Alain L’Hostis

LVMT, Univ Gustave Eiffel, IFSTTAR, ENPC, F-77447 Marne-la-Vallée, France
alain.lhostis@univ-eiffel.fr

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