Abstract

The famous Rogers-Shephard inequality states that, for any convex body \(K \subset \mathbb R^n\), we have a volumetric inequality \(V_n(K-K) \leq \binom{2n}{n}V_n(K)\) that compares the volume of the difference body of $K$, \(K-K\), with the volume of $K$. Using Cauchy's surface area formula, a particular case of the more general {\it{Kubota's Formulae for Quermassintegrals}}, we extend this classical inequality to extrapolate an upper bound \(C_K=\frac{S(K-K)}{S(K)}\) for the surface area of the difference body within the Euclidean space \(\mathbb R^n\). We accompany this upper bound with a lower bound that we derive from the classical Brunn-Minkowski inequality, \(V_n(A+B)^{1/n} \geq V_n(A)^{1/n}+V_n(B)^{1/n}\). Embracing a geometric perspective, we delve into the nuanced relationships between convex bodies and their respective surface areas, scrutinizing the patterns and properties of the difference body. This includes the validation of the upper bound in \(\mathbb R^3\) for certain classes of convex bodies, including smooth bodies and polytopes. We then analyse and establish a pattern for the formation of the difference body of pyramidal structures in \(\mathbb R^3\). Finally, we draw a conclusion on the effect of symmetry on \(C_K\)'s proximity to either bound. More specifically, we observe how deviations from symmetry, mainly when \(K\) is a simplex, often considered the most asymmetric convex body, draws the constant closer to the upper bound.