  We show that a nontrivial graph isomorphism problem of two undirected graphs,
and more generally, the permutation similarity of two given $n\times n$
matrices, is equivalent to equalities of volumes of the induced three convex
bounded polytopes intersected with a given sequence of balls, centered at the
origin with radii $t_i\in (0,\sqrt{n-1})$, where $\{t_i\}$ is an increasing
sequence converging to $\sqrt{n-1}$. These polytopes are characterized by $n^2$
inequalities in at most $n^2$ variables. The existence of fpras for computing
volumes of convex bodies gives rise to a semi-frpas of order $O^*(n^{14})$ at
most to find if given two undirected graphs are isomorphic.
