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Two Volume Product Inequalities and Their Applications

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 $\delta (K)$ such that for any $\delta \in (0, \delta(K))$ we have $\Volu(K_{\delta})\cdot \Volu((K_{\delta})^{\sstar}) \geq \Volu(K)\cdot \Volu(K^{\sstar}) \geq \Volu(K^{\delta})\cdot \Volu((K^{\delta})^{\sstar})$, where $K_{\delta}$, $K^{\delta}$ and $K^{\sstar}$ stand for the convex floating body, the illumination body, and the polar of $K$, respectively. We derive a few consequences of these inequalities.
 MSC Classifications: 52A40 - Inequalities and extremum problems 52A38 - Length, area, volume [See also 26B15, 28A75, 49Q20] 52A20 - Convex sets in $n$ dimensions (including convex hypersurfaces) [See also 53A07, 53C45]