Many villages vie for the title of France's geographical center. In each case, this means its center of gravity. But we could adopt a different definition: the point "as far as possible" from any foreign country, from which the shortest distance to the border—or the sea—is as great as possible. This notion raises some delightful mathematical problems: how to define it, whether such a point exists and how to find it!
Let us clarify the definitions!
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When we speak of the "distance to the border," we mean the minimum distance. This is therefore a problem of maximizing a minimum, just the sort of thing mathematicians relish. For simplicity, let us work in the plane, using straight-line distance—the usual distance (Euclidean distance to mathematicians). Before X becomes the border of a country, let it be any non-empty closed subset of the plane, allowing us to examine some simple cases first.
The distance r (M) from a point M in the plane to X is the infimum of the distances from M to the points of X. In practical terms, we obtain it by gradually expanding circles centered at M until they meet X: the closed disk centered at M with radius r (M) must contain points of X, but only on its boundary, called the largest empty circle centered at M.
We are therefore looking for local maxima of the distance function: points M that locally maximize r (M) for X. As M moves away from such a point M0, its distance from X decreases; in other words, the closed disk centered at M with radius r (M0) meets X.
