Optimal constants in normed planes

TL;DR

This paper introduces new geometric constants in normed planes, finds their optimal values, and explores their relationships with areas, distances, and orthogonality, providing insights into the geometry of these spaces.

Contribution

It defines and determines optimal values for new geometric constants in normed planes and characterizes the planes where these values are achieved.

Findings

New geometric constants are introduced and their optimal values are established.

Relations between these constants and areas, distances, and orthogonality are analyzed.

A conjecture on isosceles orthogonal vectors is disproved.

Abstract

We define new geometric constants for normed planes, determine their optimal values, and characterize types of planes for which these optimal values are attained. Relations of these constants to several topics, such as areas and distances from points to sides of triangles, are also investigated. We perform some calculations of new and old constants given by trigonometric functions for certain classes of norms, and a conjecture on vectors which are isosceles orthogonal is disproved.