# Properties of Minimal Distance

Triangle of Distances, Joint Analysis of Several Minimal Distances, and Measuring Inconsistency

## Abstract

The minimal distance m $$\bar D$$(G, J) between two-dimensional geosets gG and gJ has important properties, analogous to those of the minimal distance between a pair of one-dimensional sets. Napolitano (1988, 1990) demonstrated that m$$\bar D$$(G, J) is a metric, which implies that
1. 1.

m$$\bar D$$(G, J) is never negative.

2. 2.

m$$\bar D$$(G, J) = 0 if, and only if, the two geosets gG and gJ are equal, so that m$$\bar D$$(G, J) = m$$\bar D$$(G, G) = m$$\bar D$$(J, J); conversely, if m$$\bar D$$(G, J) = 0 then we can conclude that the two geosets are coincident.

3. 3.

m$$\bar D$$(G, J) = m$$\bar D$$(J, G): when we refer to the simple scheme of shifting elements of one of two geosets in order to make the two geosets coincident, this equality implies that it does not matter whether elements of gG are shifted to make gG coincident with gJ, or elements of gJ are shifted to make gJ coincident with gG.

4. 4.

If we consider three geosets gG, gJ, and gK and the minimal distance between each pair of them, we find that m$$\bar D$$(G, K) + m$$\bar D$$(K, J) ⩾ m$$\bar D$$(G, J) (14.1 a) m$$\bar D$$(G, J) + m$$\bar D$$(J, K) ⩾ m$$\bar D$$(G, K) (14.1b) m$$\bar D$$(K, G) + m$$\bar D$$(G, J) ⩾ m$$\bar D$$(K, J) (14.1c)

## Keywords

Economic Activity Minimal Distance Retail Trade Common Determinant Unitary Triangle
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