The Busemann-Petty problem on entropy of log-concave functions

Abstract

The Busemann-Petty problem asks whether symmetric convex bodies in the Euclidean space ℝnwith smaller central hyperplane sections necessarily have smaller volumes. The solution has been completed and the answer is affirmative if n ⩽ 4 and negative if n ⩾ 5. In this paper, we investigate the Busemann-Petty problem on entropy of log-concave functions: for even log-concave functions f and g with finite positive integrals in ℝn, if the marginal $$\int_{{\mathbb{R}^n} \cap H} {f(x)dx} $$ of f is smaller than the marginal $$\int_{{\mathbb{R}^n} \cap H} {g(x)dx} $$ of g for every hyperplane H passing through the origin, is the entropy Ent(f) of f bigger than the entropy Ent(g) of g? The Busemann-Petty problem on entropy of log-concave functions includes the Busemann-Petty problem, and hence its answer is negative when n ⩾ 5. For 2 ⩽ n ⩽ 4, we give a positive answer to the Busemann-Petty problem on entropy of log-concave functions.