  We show that the volume of a convex body in ${\bf R}^{n}$ in the general
membership oracle model can be computed to within relative error $\varepsilon$
using $\widetilde{O}(n^{3}\psi^{2} + n^{3}/\varepsilon^{2})$ oracle queries,
where $\psi$ is the KLS constant. With the current bound of
$\psi=\widetilde{O}(1)$, this gives an $\widetilde{O}(n^{3}/\varepsilon^{2})$
algorithm, improving on the Lov\'{a}sz-Vempala
$\widetilde{O}(n^{4}/\varepsilon^{2})$ algorithm from 2003. The main new
ingredient is an $\widetilde{O}(n^{3}\psi^{2})$ algorithm for isotropic
transformation, following which we can apply the
$\widetilde{O}(n^{3}/\varepsilon^{2})$ volume algorithm of Cousins and Vempala
for well-rounded convex bodies. We also give an efficient implementation of the
new algorithm for convex polytopes defined by $m$ inequalities in ${\bf
R}^{n}$: polytope volume can be estimated in time
$\widetilde{O}(mn^{c}/\varepsilon^{2})$ where $c<3.2$ depends on the current
matrix multiplication exponent; this improves known bounds.
