Abstract

Our main aim is to generalize the mixed quermassintegrals Wi(K,L)W_i(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)W_{\varphi,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 LpL_p-mixed quermassintegrals Wp,i(M,K,L)W_{p,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.

Author information

Contact details are reproduced from the original publication and may be historical.

Chang-Jian Zhao

Dept. of Mathematics, Jiliang University, Hangzhou 310018, P. R. China

chjzhao@aliyun.com

Keywords

  • L-p-addition
  • Orlicz addition
  • mixed quermassintegrals
  • p-mixed quermassintegrals
  • Orlicz mixed quermassintegrals
  • Orlicz projection quermassintegrals

Mathematics Subject Classification

52A20, 52A39, 46E30

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Suggested citation

C.-J. Zhao. “Inequalities for Orlicz Mixed Quermassintegrals.” Journal of Convex Analysis 26 (2019), No. 1, 129–151. https://doi.org/10.68381/jca26009

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