A Potpourri of Schwarz Related Inequalities in Inner Product Spaces

四重不等式：介于柯西-施瓦茨与三角不等式之间

Quadruple Inequalities: Between Cauchy-Schwarz and Triangle

本文证明了一族插值于柯西-施瓦茨不等式和三角不等式之间的不等式，适用于满足度量柯西-施瓦茨不等式的度量空间（如CAT(0)空间和欧氏空间），并引入了四重常数来量化函数对柯西-施瓦茨不等式的扭曲。

A class of weighted isoperimetric inequalities in hyperbolic space

In this paper, we prove a class of weighted isoperimetric inequalities for bounded domains in hyperbolic space by using the isoperimetric inequality with log-convex density in Euclidean space. As a consequence, we remove the horo-convex assumption of domains in a weighted isoperimetric inequality proved by Scheuer-Xia. Furthermore, we prove weighted isoperimetric inequalities for star-shaped domains in warped product manifolds. Particularly, we obtain a weighted isoperimetric inequality for star-shaped hypersurfaces lying outside a certain radial coordinate slice in the anti-de Sitter-Schwarzs…