Inverse Additive Problems for Minkowski Sumsets II

The Brunn-Minkowski Theorem asserts that $\mu_d(A+B)^{1/d}\geq \mu_d(A)^{1/d}+\mu_d(B)^{1/d}$ for convex bodies $A,\,B\subseteq \R^d$, where $\mu_d$ denotes the $d$-dimensional Lebesgue measure. It is well-known that equality holds if and only if $A$ and $B$ are homothetic, but few characterizations of equality in other related bounds are known. Let $H$ be a hyperplane. Bonnesen later strengthened this bound by showing $$\mu_d(A+B)\geq (M^{1/(d-1)}+N^{1/(d-1)})^{d-1}(\frac{\mu_d(A)}{M}+\frac{\mu_d(B)}{N}),$$ where $M=\sup\{\mu_{d-1}((\mathbf x+H)\cap A)\mid \mathbf x\in \R^d\}$ and $N=\sup\{\mu_{d-1}((\mathbf y+H)\cap B)\mid \mathbf y\in \R^d\}$. Standard compression arguments show that the above bound also holds when $M=\mu_{d-1}(\pi(A))$ and $N=\mu_{d-1}(\pi(B))$, where $\pi$ denotes a projection of $\mathbb R^d$ onto $H$, which gives an alternative generalization of the Brunn-Minkowski bound. In this paper, we characterize the cases of equality in this later bound, showing that equality holds if and only if $A$ and $B$ are obtained from a pair of homothetic convex bodies by `stretching' along the direction of the projection, which is made formal in the paper. When $d=2$, we characterize the case of equality in the former bound as well

On the structure of subsets of an orderable group with some small doubling properties

The aim of this paper is to present a complete description of the structure of subsets S of an orderable group G satisfying |S^2| = 3|S|-2 and is non-abelian

Small doubling in groups

Let A be a subset of a group G = (G,.). We will survey the theory of sets A with the property that |A.A| <= K|A|, where A.A = {a_1 a_2 : a_1, a_2 in A}. The case G = (Z,+) is the famous Freiman--Ruzsa theorem.Comment: 23 pages, survey article submitted to Proceedings of the Erdos Centenary conferenc

Inverse additive problems for Minkowski Sumsets II

The Brunn-Minkowski Theorem asserts that μ d (A+B) 1/d ≥μ d (A)1/d +μ d (B)1/d for convex bodies A,B⊂R d, where μ d denotes the d-dimensional Lebesgue measure. It is well known that equality holds if and only if A and B are homothetic, but few characterizations of equality in other related bounds are known. Let H be a hyperplane. Bonnesen later strengthened this bound by showing μd (A + B)≥ (M1/(d-1) + N1/(d-1))d-1 (μd (A)/M+μd(B)/N), where M=sup {μ d-1((x+H)⊂A) x ∈ R d} and N=sup{μ d-1 (y+H)∩B)∥ y∈Rd. Standard compression arguments show that the above bound also holds when M=μ d-1(π(A)) and N=μ d-1(π(B)), where π denotes a projection of R d onto H, which gives an alternative generalization of the Brunn-Minkowski bound. In this paper, we characterize the cases of equality in this latter bound, showing that equality holds if and only if A and B are obtained from a pair of homothetic convex bodies by \u27stretching\u27 along the direction of the projection, which is made formal in the paper. When d=2, we characterize the case of equality in the former bound as well. © 2011 Mathematica Josephina, Inc

A small doubling structure theorem in a Baumslag-Solitar group

Let G denote an arbitrary group. If X is a subset of G, we define its square X^2 by X^2 = {ab | a, b ∈ X}. This paper deals with the following type of problems. Let X be a finite subset of a group G. Determine the structure of X if the following inequality holds: |X^2| ≤ α|X| + β for some small α ≥ 1 and small |β|. Such problems are called inverse problems of small doubling type. We solve a general inverse problem of small doubling type in a monoid, which is a subset of the Baumslag–Solitar group BS(1, 2). Here the Baumslag-Solitar groups BS(m, n) are two-generated groups with one relation, which are defined as follows: BS(m, n) = , where m and n are integers