  A well-known family of determinantal inequalities for mixed volumes of convex
bodies were derived by Shephard from the Alexandrov-Fenchel inequality. The
classic monograph Geometric Inequalities by Burago and Zalgaller states a
conjecture on the validity of higher-order analogues of Shephard's
inequalities, which is attributed to Fedotov. In this note we disprove
Fedotov's conjecture by showing that it contradicts the Hodge-Riemann relations
for simple convex polytopes. Along the way, we make some expository remarks on
the linear algebraic and geometric aspects of these inequalities.
