The trapezoid comparison inequality in metric spaces with curvature bounded above

Abstract

Among four points of a CAT(0) space, the planar symmetric trapezoids are configurations on which Ptolemy's and Reshetnyak's inequalities are both equalities. We show that the symmetric trapezoids remain extremal for the entire family of inequalities interpolating between the two, indexed by the nondecreasing convex functions with concave derivative, and that this function class is characterized by this property. The result is qualitatively stronger than the known quadruple inequalities for this class of functions, and recovers them with their optimal constants, which were previously known only for power functions. Moreover, we extend the analysis to CAT($κ$) spaces with $κ>0$. We derive variants of Reshetnyak's quadrilateral comparison and of Ptolemy's inequality under positive upper curvature bounds, each with the optimal constant. These two inequalities yield the trapezoid comparison inequality in CAT($κ$) spaces, where the product of the bases carries an additional constant factor compared to the $κ=0$ case. Again the constant is optimal.