@inbook{7e41fb574de74c5aac0d01e291b13f78,
title = "Shephard{\textquoteright}s Inequalities, Hodge-Riemann Relations, and a Conjecture of Fedotov",
abstract = "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{\textquoteright}s inequalities, which is attributed to Fedotov. In this note we disprove Fedotov{\textquoteright}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.",
keywords = "Alexandrov-Fenchel inequality, Hodge-Riemann relations for convex polytopes, Mixed volumes, Shephard{\textquoteright}s inequalities",
author = "{van Handel}, Ramon",
note = "Publisher Copyright: {\textcopyright} 2023, The Author(s), under exclusive license to Springer Nature Switzerland AG.",
year = "2023",
doi = "10.1007/978-3-031-26300-2_13",
language = "English (US)",
series = "Lecture Notes in Mathematics",
publisher = "Springer Science and Business Media Deutschland GmbH",
pages = "337--354",
booktitle = "Lecture Notes in Mathematics",
address = "Germany",
}