Inverse Additive Problems for Minkowski Sumsets I

TL;DR

This paper characterizes the structure of extremal finite sets in the plane for certain additive inequalities involving Minkowski sums, providing equality cases and extending to projection-based inequalities.

Contribution

It introduces a detailed structural characterization of extremal sets for a key additive inequality and extends the results to projection-based inequalities.

Findings

Characterization of extremal sets for the inequality involving |A+B|

Equality cases for the bounds when m,n ≥ 2

Structural description of extremal sets in the plane

Abstract

We give the structure of discrete two-dimensional finite sets $A,\,B\subseteq \R^2$ which are extremal for the recently obtained inequality $|A+B|\ge (\frac{|A|}{m}+\frac{|B|}{n}-1)(m+n-1)$, where $m$ and $n$ are the minimum number of parallel lines covering $A$ and $B$ respectively. Via compression techniques, the above bound also holds when $m$ is the maximal number of points of $A$ contained in one of the parallel lines covering $A$ and $n$ is the maximal number of points of $B$ contained in one of the parallel lines covering $B$. When $m,\,n\geq 2$, we are able to characterize the case of equality in this bound as well. We also give the structure of extremal sets in the plane for the projection version of Bonnesen's sharpening of the Brunn-Minkowski inequality: $\mu (A+B)\ge (\mu(A)/m+\mu(B)/n)(m+n)$, where $m$ and $n$ are the lengths of the projections of $A$ and $B$ onto a line.