Applications of Gr\"unbaum-type inequalities

Abstract

Let $1\leq i \leq k < n$ be integers. We prove the following exact inequalities for any convex body $K\subset\mathbb{R}^n$ with centroid at the origin, and any $k$-dimensional subspace $E\subset \mathbb{R}^n$: \begin{align*} &V_i \big( K\cap E \big) \geq \left( \frac{i+1}{n+1} \right)^i \max_{x\in K} V_i \big( ( K-x) \cap E \big) , \\ &\widetilde{V}_i \big( K\cap E \big) \geq \left( \frac{i+1}{n+1} \right)^i \max_{x\in K} \widetilde{V}_i \big( ( K-x) \cap E \big) ; \end{align*} $V_i$ is the $i$th intrinsic volume, and $\widetilde{V}_i$ is the $i$th dual volume taken within $E$. Our results are an extension of an inequality of M. Fradelizi, which corresponds to the case $i=k$. Using the same techniques, we also establish extensions of "Gr\"unbaum's inequality for sections" and "Gr\"unbaum's inequality for projections" to dual volumes.