A Generalization of a Classical Geometric Extremum Problem

TL;DR

This paper generalizes classical geometric extremum problems by analyzing shortest and longest segments through an interior point of convex bodies, revealing smoothness and supporting hyperplane properties, with extensions to polytopes and external points.

Contribution

It introduces new geometric characterizations of extremal segments in convex bodies, including smoothness and hyperplane intersection properties, extending classical results to higher dimensions and various convex shapes.

Findings

Shortest segments imply boundary smoothness at endpoints.

Normals at endpoints intersect at a point related to the interior point.

Results extend to convex polytopes and external points.

Abstract

Let $\partial \,\mathcal{C}$ be the boundary of a compact convex body $\mathcal{C}$ in $\mathbb{R}^n,\, n\geq 2$, and $O$ be an interior point of $\mathcal C$. Every straight line $l$ containing $O$ cuts from $\mathcal{C}$ a segment $[AB]$ with end-points on $\partial \,\mathcal{C}$. It is shown that if $[AB]$ is the shortest such segment, then $\partial \,\mathcal{C}$ is smooth at the points $A$ and $ B$ (i.e. at both of them there is only one supporting hyperplane for $\mathcal{C}$) and, something more, the normals to the unique supporting hyperplanes at the points $A$ and $B$ intersect at a point belonging to the hiperplane through $O$ which is orthogonal to $[AB]$. If $\mathcal{C}$ is a smooth compact convex body in $\mathbb{R}^n,\, n\geq 2$, the above property holds also when $[AB]$ is the longest such segment. Similar results have place also when $O$ is outside the set $\mathcal{C}$. The ``local versions'' of these results (when the length $|AB|$ of the segment $[AB]$ is locally maximal or locally minimal) also have a place. More specific results are obtained in the particular case when $\mathcal{C}$ is a convex polytope.