Title: New fiber and graph combinations of convex bodies Show Chinese title

Title: 凸体的新纤维和图组合

Abstract: Three new combinations of convex bodies are introduced and studied: the $L_p$ fiber, $L_p$ chord and graph combinations. These combinations are defined in terms of the fibers and graphs of pairs of convex bodies, and each operation generalizes the classical Steiner symmetral, albeit in different ways. For the $L_p$ fiber and $L_p$ chord combinations, we derive Brunn--Minkowski-type inequalities and the corresponding Minkowski's first inequalities. We also prove that the general affine surface areas are concave (respectively, convex) with respect to the graph sum, thereby generalizing fundamental results of Ye (Indiana Univ. Math. J., 2014) on the monotonicity of the general affine surface areas under Steiner symmetrization. As an application, we deduce a corresponding Minkowski's first inequality for the $L_p$ affine surface area of a graph combination of convex bodies. Show Chinese abstract

Abstract: 引入并研究了三种新的凸体组合：$L_p$纤维、$L_p$弦和图组合。 这些组合是通过凸体对的纤维和图来定义的，每种运算都以不同的方式推广了经典的Steiner对称体。 对于$L_p$纤维和$L_p$弦组合，我们推导出了Brunn--Minkowski型不等式和相应的Minkowski第一不等式。 我们还证明了广义仿射面积相对于图和是凹的（分别凸的），从而推广了Ye在Steiner对称化下广义仿射面积单调性的基本结果（Indiana Univ. Math. J., 2014）。 作为应用，我们推导出了凸体图组合的$L_p$仿射面积相应的Minkowski第一不等式。