  Many classical geometric inequalities on functionals of convex bodies depend
on the dimension of the ambient space. We show that this dimension dependence
may often be replaced (totally or partially) by different symmetry measures of
the convex body. Since these coefficients are bounded by the dimension but
possibly smaller, our inequalities sharpen the original ones. Since they can
often be computed efficiently, the improved bounds may also be used to obtain
better bounds in approximation algorithms.
