Slicing convex sets and measures by a hyperplane

TL;DR

This paper extends the ham sandwich theorem to well-separated measures and convex bodies, providing conditions for the existence and uniqueness of a hyperplane that slices each body or measure proportionally.

Contribution

It generalizes the ham sandwich theorem to well-separated measures and convex bodies, establishing conditions for a hyperplane to bisect multiple objects proportionally.

Findings

Established sufficient conditions for hyperplanes to slice convex bodies proportionally.

Extended the result from convex bodies to measures.

Proved uniqueness of such hyperplanes under certain conditions.

Abstract

We generalize the ham sandwich theorem for the case of well separated measures. Given convex bodies $K_1,...,K_d$ in $\mathbb{R_d}$ and numbers $\alpha_1,...,\alpha_d \in [0, 1]$, we give a sufficient condition for existence and uniqueness of an (oriented)halfspace H with Vol($H \cap K_i$)= $\alpha_i \dot$ Vol$K_i$ for every i. The result is extended from convex bodies to measures.