Anti-concentration inequalities for log-concave variables on the real line

TL;DR

This paper establishes precise anti-concentration bounds for log-concave random variables on the real line, applicable to both discrete and continuous cases, using elementary majorization methods.

Contribution

It introduces a simple, elementary approach to derive sharp anti-concentration inequalities for log-concave variables, extending existing results.

Findings

Sharp anti-concentration bounds for log-concave variables

Unified approach for discrete and continuous cases

Extension of recent and classical results

Abstract

We prove sharp anti-concentration results for log-concave random variables on the real line in both the discrete and continuous setting. Our approach is elementary and uses majorization techniques to recover and extend some recent and not so recent results.