Generalized Grünbaum inequality

Abstract

Let $f$ be an integrable log-concave function on ${\mathbb{R}}^n$ with the center of mass at the origin. We show that $\underset{0}{\overset{\infty }{\int }}f(s\theta )ds\ge e^{-n}\underset{-\infty }{\overset{\infty }{\int }}f(s\theta )ds$ for every $\theta \in S^{n-1}$, and the constant $e^{-n}$ is the best possible.

8 pages, 4 figures