Mixed volumes and the Bochner method

Abstract

At the heart of convex geometry lies the observation that the volume of convex bodies behaves as a polynomial. Many geometric inequalities may be expressed in terms of the coefficients of this polynomial, called mixed volumes. Among the deepest results of this theory is the Alexandrov-Fenchel inequality, which subsumes many known inequalities as special cases. The aim of this note is to give new proofs of the Alexandrov-Fenchel inequality and of its matrix counterpart, Alexandrov's inequality for mixed discriminants, that appear conceptually and technically simpler than earlier proofs and clarify the underlying structure. Our main observation is that these inequalities can be reduced by the spectral theorem to certain trivial "Bochner formulas".

One more proof of the Alexandrov–Fenchel inequality

“…This short text has the modest goal of presenting another proof of the Alexandrov-Fenchel inequality for mixed volumes. It is similar to the proof for polytopes discussed in the recent work of Shenfeld and van Handel [5] which is mainly devoted to a short and neat treatment of the case of smooth bodies. Our presentation puts the emphasis on the basic algebraic properties of the polynomials involved in the construction of mixed volumes.…”

“…The history of the Alexandrov-Fenchel inequality and its various proofs until the 1980s is described in the book by Burago and Zalgaller [1,Section 20.3]. The more recent literature contains a proof by Wang [7] that was inspired by Gromov's work [2], in addition to the proof by Shenfeld and van Handel [5]. Applications to combinatorics are described by Stanley [6].…”

The local logarithmic Brunn-Minkowski inequality for zonoids

“…A new approach to the study of the spectral properties of A K was discovered by Shenfeld and the author in [20]. This approach, called the Bochner method in view of its analogy to the classical Bochner method in differential geometry, has found several surprising applications both inside and outside convex geometry.…”

“…This approach, called the Bochner method in view of its analogy to the classical Bochner method in differential geometry, has found several surprising applications both inside and outside convex geometry. The Bochner method was already used in [20] to provide new proofs of the Alexandrov-Fenchel inequality, a much deeper result of which (1.6) is a special case, and of the Alexandrov mixed discriminant inequality. Subsequent applications outside convexity include the proof of certain properties of Lorentzian polynomials in [9,5] and the striking results of [6], where the method is used to prove numerous combinatorial inequalities.…”

On Hodge-Riemann relations for translation-invariant valuations

“…To conclude the introduction, let us point out that among the numerous known proofs of the Aleksandrov-Fenchel inequality [2,3,27,42,46,51] (see also [26], §27), one is particularly relevant to our approach. Namely, McMullen [42] regards the inequality as a special case of the (mixed) Hodge-Riemann relations in his polytope algebra [41].…”