A Functional Busemann Intersection Inequality
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A Functional Busemann Intersection Inequality Songjun Lv1,2 Received: 5 February 2020 / Accepted: 21 September 2020 © Mathematica Josephina, Inc. 2020
Abstract The Busemann intersection inequality states that if K is a compact domain in Rn then vol(K ∩ u ⊥ )n du, vol(K )n−1 ≥ c(n) S n−1
where c(n) > 0 is an explicit constant, with equality if and only if K is an ellipsoid centered at the origin. In this paper, we prove a functional version of the Busemann intersection inequality. We also demonstrate an “equivalent” sharp entropy inequality for dual mixed volumes of functions. Keywords Intersection body · Busemann intersection inequality · Dual Minkowski inequality · Contoured function Mathematics Subject Classification 26D · 52A
1 Introduction Affine functional inequalities, which turn out to be functional inequalities with an intense geometric background, have attracted increasing interest in recent decades. Basic connections between functional inequalities and geometric counterparts are those between the classical Sobolev inequality and the isoperimetric inequality in the n-Euclidean space Rn , and between the Prékopa-Leindler inequality and the BrunnMinkowski inequality. In 1999, Zhang [29] creatively formulated the affine Sobolev inequality by integrating the Petty projection inequality into the Sobolev space theory.
Research supported partly by Natural Science Foundation of Chongqing China under Grants cstc2018jcyjAX0190.
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Songjun Lv [email protected]
1
School of Mathematics and Statistics, Changshu Institute of Technology, Suzhou 215500, China
2
School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, China
123
S. Lv
He also demonstrated that this affine functional inequality is much stronger than its Euclidean counterpart. Surprisingly, the affine Sobolev–Zhang inequality shows that the affine Sobolev theory may not rely strongly on the Euclidean geometric structure of Rn as its Euclidean counterpart does. In 2002, Lutwak et al. [22] used a quite different approach to disclose a strong connection between the so-called affine L p Sobolev inequality and the L p Petty projection inequality [21] for p > 1. Since then, a number of affine functional inequalities [2,11–14,17,23–28] have been demonstrated and now constitute the main components of the affine functional inequality theory. In this paper, we follow the exquisite line mentioned above to continue building up substantial connections between the theory of functional inequalities and convex geometry. We shall demonstrate a new affine functional inequality, whose geometric counterpart is the Busemann intersection inequality from convex geometry. Observing that most of the affine invariants appearing in known affine functional inequalities are in terms of p-cosine transforms of various geometric measures, we begin with a limiting case of the p-cosine transforms of given C 1 functions. For a nonnegative C 1 function f with compact support in Rn , the p-cosine transform of f can be expressed as C p f (ξ ) =
Rn
|x · ξ |
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