  We prove the following theorem. Let $\mu$ be a measure on $R^n$ with even
continuous density, and let $K,L$ be origin-symmetric convex bodies in $R^n$ so
that $\mu(K\cap H)\le \mu(L\cap H)$ for any central hyperplane H. Then
$\mu(K)\le \sqrt{n} \mu(L).$ We also prove this result with better constants
for some special classes of measures and bodies. Finally, we prove a version of
the hyperplane inequality for convex measures.
