Mahler-type volume inequality for convex bodies with tetrahedral symmetry

Let $K$ be a convex body in ${\mathbb{R}}^n$. We denote the volume of $K$ by $\mathrm{\mid }K\mathrm{\mid }$, and the polar body of its difference body $K-K$ by $(K-K{)}^{\circ }$. We provide a new proof of the well-known estimate $$\mathrm{\mid }K\mathrm{\mid }\mathrm{\mid }(K-K{)}^{\circ }\mathrm{\mid }\ge \frac{3}{2}$$ for $K\subset {\mathbb{R}}^2$, with equality attained for a triangle. For $K\subset {\mathbb{R}}^3$ with tetrahedral symmetry, we prove that $$\mathrm{\mid }K\mathrm{\mid }\mathrm{\mid }(K-K{)}^{\circ }\mathrm{\mid }\ge \frac{2}{3},$$ with equality attained for a tetrahedron.