Triangles Inscribed in a Semicircle, in Minkowski Planes, and in Normed Spaces

Abstract

In this paper we mainly consider triangles inscribed in a semicircle of a normed space; in two-dimensional spaces, their perimeter has connections with the perimeter of the sphere. Moreover, by using the largest values the perimeter of such triangles can have, we define two new, simple parameters in real normed spaces: one of these parameters is strictly connected with the modulus of convexity of the space, while the study of the other one seems to be more complicated. We calculate the value of our two parameters and we bring out a few connections among their values and the geometry of real normed spaces.