Kadets type theorems for partitions of a convex body

TL;DR

This paper extends Kadets type theorems to convex partitions, showing that in the plane any partition allows a homothetic copy of the original convex body with a sum of coefficients at least one, with restrictions in higher dimensions.

Contribution

It generalizes Kadets type theorems to convex partitions, providing new bounds on homothety coefficients in various dimensions.

Findings

In the plane, any convex partition admits a homothetic copy with sum of coefficients ≥ 1.

In higher dimensions, additional restrictions on partitions are necessary.

The results unify and extend classical geometric theorems on convex body partitions.

Abstract

For convex partitions of a convex body $B$ we prove that we can put a homothetic copy of $B$ into each set of the partition so that the sum of homothety coefficients is $\ge 1$. In the plane the partition may be arbitrary, while in higher dimensions we need certain restrictions on the partition.