Generalized Gr\"unbaum inequality

TL;DR

This paper extends the Grünbaum inequality to integrable log-concave functions in n-dimensional space, establishing a sharp lower bound for integrals over rays emanating from the origin in any direction.

Contribution

It generalizes the Grünbaum inequality from convex bodies to log-concave functions, providing the best possible constant in this broader setting.

Findings

Established a lower bound for integrals of log-concave functions along rays.

Proved the constant e^{-n} is optimal for the inequality.

Extended classical geometric inequalities to a functional setting.

Abstract

Let $f$ be an integrable log-concave function on ${\mathbb R^n}$ with the center of mass at the origin. We show that $\int\limits_0^{\infty}f(s\theta)ds\ge e^{-n}\int\limits_{-\infty}^{\infty}f(s\theta)ds$ for every $ \theta\in S^{n-1}$, and the constant $e^{-n}$ is the best possible.