Two Volume Product Inequalities and Their Applications

Stancu, Alina

Canadian mathematical bulletin, Tome 52 (2009) no. 3, pp. 464-472

Résumé

Let $K\,\subset \,{{\mathbb{R}}^{n+1}}$ be a convex body of class ${{C}^{2}}$ with everywhere positive Gauss curvature. We show that there exists a positive number $\delta \left( K \right)$ such that for any $\delta \,\in \,\left( 0,\,\delta \left( K \right) \right)$ we have $\text{Vol}\left( {{K}_{\delta }} \right)\,\cdot \,\text{Vol}\left( {{\left( {{K}_{\delta }} \right)}^{*}} \right)\,\ge \,\text{Vol}\left( K \right)\,\cdot \,\text{Vol}\left( {{K}^{*}} \right)\,\ge \,\text{Vol}\left( {{K}^{\delta }} \right)\,\cdot \,\text{Vol}\left( {{\left( {{K}^{\delta }} \right)}^{*}} \right)$ , where ${{K}_{\delta }}$ , ${{K}^{\delta }}$ and ${{K}^{*}}$ stand for the convex floating body, the illumination body, and the polar of $K$ , respectively. We derive a few consequences of these inequalities.

Détail

DOI : 10.4153/CMB-2009-049-0

Mots-clés : 52A40, 52A38, 52A20, affine invariants, convex floating bodies, illumination bodies