Minimum radius balls in metric spaces with the Heine-Borel property, including Hilbert, Thompson, and Funk geometries, are LP-type problems.
Flag-approximability of convex bodies and volume growth of Hilbert geometries
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abstract
We introduce the flag-approximability of a convex body to measure how easy it is to approximate by polytopes. We show that the flag-approximability is exactly half the volume entropy of the Hilbert geometry on the body, and that both quantities are maximized when the convex body is a Euclidean ball. We also compute explicitly the asymptotic volume of a convex polytope, which allows us to prove that simplices have the least asymptotic volume.
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2024 1verdicts
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On The Heine-Borel Property and Minimum Enclosing Balls
Minimum radius balls in metric spaces with the Heine-Borel property, including Hilbert, Thompson, and Funk geometries, are LP-type problems.