Journal of Convex Analysis 34 (2027), No. 1, 043--066
Copyright Heldermann Verlag 2027
A Generalization of a Classical Geometric Extremum Problem
Petar Kenderov
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria
vorednek@gmail.com
Oleg Mushkarov
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria
muskarov@math.bas.bg
Nikolai Nikolov
(1) Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria
(2) Faculty of Information Sciences, State University of Library Studies and Information Technologies, Sofia, Bulgaria
nik@math.bas.bg
[Abstract-pdf]
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 hyperplane through $O$ which is orthogonal to $[AB]$.\\[1mm] 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 are also valid 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) are valid as well. More specific results are obtained in the particular case when $\mathcal{C}$ is a convex polytope.
Keywords: Convex set, extremal problem, Philo's line, critical point, convex polytope, concurrent lines.
MSC: 51M16, 52A20, 52A40, 49K05, 49K10.
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