Abstract
Böröczky et al. proposed the log-Minkowski problem and established the plane log-Minkowski inequality for origin-symmetric convex bodies. Recently, Stancu proved the log-Minkowski inequality for mixed volumes; Wang, Xu, and Zhou gave the L_{p} version of Stancu’s results. In this paper, we define the L_{p}-mixed quermassintegrals probability measure and obtain the log-Minkowski inequality for the L_{p}-mixed quermassintegrals. As its application, we establish the L_{p}-mixed affine isoperimetric inequality. In addition, we also consider the dual log-Minkowski inequalities for the L_{p}-dual mixed quermassintegrals.
Highlights
Introduction and main results LetKn denote a set of convex bodies in Euclidean space Rn
For the set of convex bodies containing the origin in their interiors and the set of origin-symmetric convex bodies in Rn, we write Kon and Kons, respectively
In relation to Lp-mixed quermassintegrals, we continuously study logMinkowski inequality
Summary
+ V (L) n , with equality if and only if K and L are homothetic. Here K + L = {x + y : x ∈ K and y ∈ L} denotes the Minkowski sum of K and L. Theorem 1.C (The log-Minkowski inequality for Lp-mixed volume) If K, L ∈ Kon, p > 1, ln hK dV p(L, K ). According to (1.14) and (1.15), we define the Lp-mixed quermassintegral probability measure as follows: For K, L ∈ Kon, p ≥ 1, and i = 0, 1, . Combined with the above Lp-mixed quermassintegral probability measure, we give a generalization of the log-Minkowski inequalities (1.6) and (1.13). Theorem 1.1 (The log-Minkowski inequality for Lp-mixed quermassintegral) If K, L ∈ Kon, p ≥ 1, and i = 0, 1, 2, . Combined with (1.16), (1.11), and (1.12), we have ln dW i(L) ≥ ln
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