City shapes that maximize the number of walking-only trips based on Manhattan distance

Various modes of transportation are available when people travel within cities, and trips can be classified into two types depending on whether some type of vehicle is used. Compared to vehicular travel, trips conducted only by walking have the advantages of lower environmental impact and less space required for road networks. By assuming that the proportion of walking-only trips decreases exponentially with the distance traveled, we explore the problem of finding a city shape with a fixed land area that maximizes the number of walking-only trips based on Manhattan distance. For many-to-one travel with the city center as the destination, we show that the optimal city shape is a diamond. For many-to-many travel, a method is presented that expresses the number of walking-only trips as a double integral, originally formulated as a four-dimensional integral. Using this, an optimization problem is formulated whose variables are the vertex coordinates of a polygon, and approximate solutions for the optimal city shape under several different settings are obtained numerically. For many-to-many travel, it is shown that a large number of walking-only trips occur when the city shape is close to being circular, although the exact shape varies with the distance deterrence coefficient.