Gurvits’s conjecture on the impossibility of an FPRAS for general mixed volumes

Prove the nonexistence of a Fully-Polynomial Randomized Approximation Scheme (FPRAS) for approximating general mixed volumes within the Dyer–Gritzmann–Hufnagel oracle model for well-presented convex bodies, i.e., establish that no algorithm can approximate mixed volumes to within a multiplicative factor (1±ε) in time polynomial in input size and ε^{-1} under that model.

Background

The paper studies randomized polynomial-time approximation of mixed volumes for tuples of lattice polytopes, providing positive results when the number of bodies k is constant and the polytopes are given by convex hulls of polynomially many lattice points. In contrast, prior work by Dyer–Gritzmann–Hufnagel showed #P-hardness for computing mixed volumes in an oracle model for well-presented convex bodies, highlighting significant barriers for general instances.

Within this broader complexity landscape, Gurvits formulated a conjecture asserting that, in the Dyer–Gritzmann–Hufnagel oracle model, it is impossible to design an FPRAS for general mixed volumes. This conjecture delineates a fundamental open question about the approximability of mixed volumes beyond the specific settings treated by the current paper.