On the analogue of the monotonicity of entropy in the Brunn-Minkowski theory

Series: 
Stochastics Seminar
Thursday, March 3, 2016 - 15:05
1 hour (actually 50 minutes)
Location: 
Skiles 006
,  
IMA, University of Minnesota
Organizer: 
In the late 80's, several relationships have been established between the Information Theory and Convex Geometry, notably through the pioneering work of Costa, Cover, Dembo and Thomas. In this talk, we will focus on one particular relationship. More precisely, we will focus on the following conjecture of Bobkov, Madiman, and Wang (2011), seen as the analogue of the monotonicity of entropy in the Brunn-Minkowski theory: The inequality $$ |A_1 + \cdots + A_k|^{1/n} \geq \frac{1}{k-1} \sum_{i=1}^k |\sum_{j \in \{1, \dots, k\} \setminus \{i\}} A_j |^{1/n}, $$ holds for every compact sets $A_1, \dots, A_k \subset \mathbb{R}^n$. Here, $|\cdot|$ denotes Lebesgue measure in $\mathbb{R}^n$ and $A + B = \{a+b : a \in A, b \in B \}$ denotes the Minkowski sum of $A$ and $B$. (Based on a joint work with M. Fradelizi, M. Madiman, and A. Zvavitch.)