Inequalities for Orlicz Mixed Quermassintegrals

Our main aim is to generalize the mixed quermassintegrals $W_i(K, L)$ of convex bodies to the Orlicz space. Under the framework of Orlicz-Brunn-Minkowski theory, we introduce a new affine geometric quantity $W_{\varphi,i}(M, K, L)$ 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 $L_p$-mixed quermassintegrals $W_{p,i}(M, K, L)$ 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.