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Journal of Convex Analysis 26 (2019), No. 1, [final page numbers not yet available]
Copyright Heldermann Verlag 2019



Inequalities for Orlicz Mixed Quermassintegrals

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



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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