A generalization of Winternitz’s theorem and its discrete version

Shyntar, Alexandra; Yaskin, Vladyslav

doi:10.1090/proc/15465

A generalization of Winternitz’s theorem and its discrete version
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by Alexandra Shyntar and Vladyslav Yaskin PDF

Proc. Amer. Math. Soc. 149 (2021), 3089-3104 Request permission

Abstract:

Let $K$ be a convex body in the plane. Cut $K$ by a line passing through its centroid. It is a well-known result, due to Winternitz, that the areas of the resulting two pieces are at least $4/9$ times the area of $K$ and at most $5/9$ times the area of $K$. We generalize this inequality to the case when the body is cut by a line not passing through the centroid. As an application we obtain a discrete version of Winternitz’s theorem.

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