A Volume Product Representation and its Ramifications in lp,

TL;DR

This paper introduces a new representation of the volume product for the unit p-ball that simplifies analysis and confirms classical bounds using basic gamma function theory and classical analysis.

Contribution

It provides a gamma-symbolism-free representation of the volume product, enabling simpler proofs of Mahler's lower bound and Santaló's upper bound.

Findings

Confirmed Mahler's lower bound using classical analysis.

Confirmed Santaló's upper bound with gamma function theory.

Presented a novel, simplified representation of the volume product.

Abstract

We represent the volume product for the unit p-ball in a a form free of its gamma symbolism;this will enable us to confirm Mahler's lower bound and Santalo's upper bound by the use of basic only gamma function theory and moderately advanced classical analysis.