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 $b\ge0$. Exact upper and lower bounds on the corresponding exponential moments (Proposition 9.4) and explicit exponentially decreasing upper bounds on the mentioned p.d.f.'s (Propositions 9.2 and 9.3) are given as well. Some general results concerning log-concave p.d.f.'s are also obtained.