SMALL BALL PROBABILITY ESTIMATES FOR LOG-CONCAVE MEASURES Academic Article

We establish a small ball probability inequality for isotropic log log -concave probability measures: there exist absolute constants c 1 , c 2 > 0 c_{1}, c_{2}>0 such that if X X is an isotropic log log -concave random vector in R n {mathbb R}^n with 2 psi _{2} constant and bounded by b b and if A A is a non-zero n n n imes n matrix, then for every ( 0 , c 1 ) varepsilon in (0,c_{1}) and y R n y in mathbb R^n , [ P ( A x