Applications of Grünbaum-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 “Grünbaum’s inequality for sections” and “Grünbaum’s inequality for projections” to dual volumes.