Discrete Brunn–Minkowski inequalities

Abstract: The Brunn–Minkowski inequality is a cornerstone of convex geometry, with deep connections to several areas of mathematics. In recent years, there has been growing interest in developing discrete versions of this inequality. Attempts to formulate a discrete version of the Brunn–Minkowski inequality naturally lead to problems in additive combinatorics, particularly those involving lower bounds and structural aspects of finite sumsets in ${\mathbb R}^d$ or ${\mathbb Z}^d$. In the continuous setting, a refinement due to Bonnesen incorporates the $(d-1)$-dimensional volume of projections onto a hyperplane, yielding sharper bounds that capture geometric structure. A discrete counterpart of this refinement is currently known only in dimension two, due to Grynkiewicz and Serra. In this paper, we explore extensions of this result to higher dimensions. In particular, we introduce a framework for deriving discrete Brunn–Minkowski-type inequalities in arbitrary dimension that incorporate projection data of the underlying sets. This is a joint work with Oriol Serra and Luis Montejano.

number theory

Audience: researchers in the topic

Combinatorial and additive number theory seminar (CANT 2026)