Dual Loomis-Whitney Inequalities via Information Theory

• Main Authors: Jing Hao, Varun Jog
• Format: Article
• Language: English
• Published: MDPI AG 2019-08-01
• Series: Entropy
• Subjects: Loomis-Whitney inequality; fisher information; volume; surface area; log-concave distributions
• Online Access: https://www.mdpi.com/1099-4300/21/8/809

We establish lower bounds on the volume and the surface area of a geometric body using the size of its slices along different directions. In the first part of the paper, we derive volume bounds for convex bodies using generalized subadditivity properties of entropy combined with entropy bounds for log-concave random variables. In the second part, we investigate a new notion of Fisher information which we call the <inline-formula> <math display="inline"> <semantics> <msub> <mi>L</mi> <mn>1</mn> </msub> </semantics> </math> </inline-formula>-Fisher information and show that certain superadditivity properties of the <inline-formula> <math display="inline"> <semantics> <msub> <mi>L</mi> <mn>1</mn> </msub> </semantics> </math> </inline-formula>-Fisher information lead to lower bounds for the surface areas of polyconvex sets in terms of its slices.