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

Shephard's inequalities, Hodge-Riemann relations, and a conjecture of Fedotov