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Journal of Convex Analysis 26 (2019), No. 1, 129--151
Copyright Heldermann Verlag 2019

Inequalities for Orlicz Mixed Quermassintegrals

Chang-Jian Zhao
Dept. of Mathematics, Jiliang University, Hangzhou 310018, P. R. China

Our main aim is to generalize the mixed quermassintegrals Wi(K, L) of convex bodies to the Orlicz space. Under the framework of Orlicz-Brunn-Minkowski theory, we introduce a new affine geometric quantity Wφ,i(M, K, L) by calculating the first Orlicz variation of the mixed quermassintegrals, and call it the Orlicz mixed quermassintegrals of the convex bodies M, K and L. Fundamental notions and properties of mixed quermassintegrals, and the Minkoswki and Brunn-Minkowski inequalities for mixed quermassintegrals are derived in the Orlicz setting. Related concepts and inequalities of a new type of Lp-mixed quermassintegrals Wp,i(M, K, L) are also derived. One of these has connections with the conjectured log-Brunn-Minkowski inequality and we prove a new general log Minkowski type inequality. Finally, we introduce the concept of mixed projection quermassintegrals and prove an Orlicz-Minkowski type inequality for the mixed projection quermassintegrals.

Keywords: L-p-addition, Orlicz addition, mixed quermassintegrals, p-mixed quermassintegrals, Orlicz mixed quermassintegrals, Orlicz projection quermassintegrals.

MSC: 52A20, 52A39, 46E30

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