On the Volume Product of the Unit Balls of Lipschitz-Free Spaces

Abstract

In this thesis, we examine the volume product of the unit balls in Lipschitz-free spaces. In particular, we study metric spaces corresponding to different graphs having 3, 4, or 5 vertices. Our analysis involves studying the structure of Lipschitz-free spaces over these graphs. We construct the unit ball in the Lipschitz-free space and find its polar body, i.e., the unit ball in the dual norm. Next, we compute the volumes of the unit ball and the polar body and find the exact value of the volume product of those spaces. We compare the outcome of this computation with the volume product of the unit cube to confirm Mahler’s conjecture for those Lipschitz-free spaces. In this thesis, we also review many essential definitions and facts before presenting examples to support our analysis. This work is based on the fundamental properties of Lipschitz functions and spaces, emphasizing their geometric and algebraic properties.