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arXiv · 1501.03907

Subdivisions of rotationally symmetric planar convex bodies minimizing the maximum relative diameter

Antonio Cañete·Uwe Schnell·Salvador Segura Gomis
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Abstract

In this work we study subdivisions of $k$-rotationally symmetric planar convex bodies that minimize the maximum relative diameter functional. For some particular subdivisions called $k$-partitions, consisting of $k$ curves meeting in an interior vertex, we prove that the so-called \emph{standard $k$-partition} (given by $k$ equiangular inradius segments) is minimizing for any $k\in\mathbb{N}$, $k\geq 3$. For general subdivisions, we show that the previous result only holds for $k\leq 6$. We also study the optimal set for this problem, obtaining that for each $k\in\mathbb{N}$, $k\geq 3$, it consists of the intersection of the unit circle with the corresponding regular $k$-gon of certain area. Finally, we also discuss the problem for planar convex sets and large values of $k$, and conjecture the optimal $k$-subdivision in this case.

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Antonio Cañete, Uwe Schnell, Salvador Segura Gomis. 2015-01-16. Subdivisions of rotationally symmetric planar convex bodies minimizing the maximum relative diameter. https://arxiv.org/abs/1501.03907

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