A critical condition for the cost density in the circular city model

In regard to the problem of determining minimal-cost routes in a region with variable cost density, it has been shown elsewhere that, for a radially symmetric cost density which is inversely proportional to the distance from a central point O, the minimal cost between two pointsP ₁ andP ₂ which are equidistant from O is attained along a circular arc. This is not true in general for an arbitrary, radially symmetric cost density. In the present paper, critical conditions for determining when a circular arc will be a relative minimal-cost path are derived. These criteria are then employed to examine a class of special cases in which the cost density is constant outside the city limits.The author would like to express his appreciation to Professor J. B. Keller for calling his attention to the work of R. K. Luneberg and to Professor K. A. Brakke for some helpful advice.