A Generalization of a Classical Geometric Extremum Problem

Let ∂ C be the boundary of a compact convex body C in Rn, n≥2, and O be an interior point of C. Every straight line l containing O cuts from C a segment [AB] with end-points on ∂ C. It is shown that if [AB] is the shortest such segment, then ∂ C is smooth at the points A and B (i.e. at both of them there is only one supporting hyperplane for C) and, something more, the normals to the unique supporting hyperplanes at the points A and B intersect at a point belonging to the hyperplane through O which is orthogonal to [AB]. [1mm] If C is a smooth compact convex body in Rn, n≥2, the above property holds also when [AB] is the longest such segment. Similar results are also valid when O is outside the set C. The “local versions” of these results (when the length ∣AB∣ of the segment [AB] is locally maximal or locally minimal) are valid as well. More specific results are obtained in the particular case when C is a convex polytope