  The Busemann-Petty problem asks whether symmetric convex bodies in the
  Euclidean space $\mathbb{R}^n$ with smaller central hyperplane sections
necessarily have smaller volume. The solution has been completed and the answer
is affirmative if $n \le 4$ and negative if $n\ge 5$. 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
$\mathbb{R}^n$, 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, whether the entropy ${\rm Ent}(f)$
of $f$ is bigger than the entropy ${\rm Ent}(g)$ of $g$? The Busemann-Petty
problem on entropy of log-concave functions includes the Busemann-Petty
problem, hence, its answer is negative when $n\geq5$. For $2\leq n\leq4$ we
give a positive answer to the Busemann-Petty problem on entropy of log-concave
functions.
