Random Translates in Minkowski Sums

TL;DR

This paper investigates how to choose sets in Euclidean space and the torus to minimize the expected volume of their sumsets when adding random points, showing that balls and bands are optimal choices.

Contribution

It proves that Euclidean balls minimize the expected sumset volume in Euclidean space, and bands are optimal in the torus, with stability results included.

Findings

Balls minimize expected sumset volume in Euclidean space.

Bands are optimal in the torus setting.

Stability versions of the optimality results are provided.

Abstract

Suppose that $A$ and $B$ are sets in $\mathbb{R}^d$, and we form the sumset of $A$ with $n$ random points of $B$. Given the volumes of $A$ and $B$, how should we choose them to minimize the expected volume of this sumset? Our aim in this paper is to show that we should take $A$ and $B$ to be Euclidean balls. We also consider the analogous question in the torus $\mathbb{T}^d$, and we show that in this case the optimal choices of $A$ and $B$ are bands, in other words, sets of the form $[x,y]\times\mathbb{T}^{d-1}$. We also give stability versions of our results.