The mixed volume 
of 
nonempty compact convex sets in
is 
times the coefficient of
in the polynomial 
, where addition
is the Minkowski sum and 
scales 
(Schneider 2014). It is symmetric in its arguments and linear
in each argument with respect to Minkowski sum and
nonnegative scaling. Its normalization satisfies
 |
Thus mixed volume extends ordinary volume by recording interactions between different convex sets.
Chen and Liu (2026) gave a fully polynomial-time randomized approximation scheme for mixed volume. For convex
bodies, the input consists of weak membership
oracles and rational numbers specifying a
center and positive inner and outer radii. The inner closed ball about that center lies in the body, which
lies in the outer closed ball. Given 
, the algorithm
returns an estimate 
with
![Pr[(1-epsilon)V<=V^^<=(1+epsilon)V]>=1-delta.](/images/equations/MixedVolume/NumberedEquation2.svg) |
Its numbers of oracle calls and bit operations are bounded by a polynomial in 
, 
,
, and the binary input length.
Their extension to lower-dimensional compact convex
sets additionally uses rational descriptions of the affine
hulls, with membership tolerances and inner and outer ball data measured relative
to those affine hulls. Chen and Liu (2026) attribute
the main mathematical ideas to GPT-6 Astra and state that they simplified and rewrote
the arguments and take responsibility for them.
