Exact bounds on the distribution function of isotropic log-concave distributions

Abstract

For each real $b$, exact upper and lower bounds on the probability $\mathsf P(X\ge b)$ over all random variables $X$ with log-concave p.d.f.'s such that $\mathsf E X=0$ and $\mathsf E X^2=1$ are obtained, as well as the best constant factor $C$ in the inequality $\mathsf P(X\ge b)\le C e^{-b}$ for all real $b\ge0$. Explicit exponentially decreasing upper bounds on the mentioned p.d.f.'s are given as well. Some general results concerning log-concave p.d.f.'s are also obtained.