  We discuss an analytic form of the dilation inequality for symmetric convex
sets in Euclidean spaces, which is a counterpart of analytic aspects of
Cheeger's isoperimetric inequality. We show that the dilation inequality for
symmetric convex sets is equivalent to a certain bound of the relative entropy
for symmetric quasi-convex functions, which is close to the logarithmic Sobolev
inequality or Cram\'{e}r--Rao inequality. As corollaries, we investigate the
reverse Shannon inequality, logarithmic Sobolev inequality, Kahane--Khintchine
inequality, deviation inequality and isoperimetry. We also give new probability
measures satisfying the dilation inequality for symmetric convex sets via
bounded perturbations and tensorization.
