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.

Let $(X, \norm)$ be a Minkowski space (finite dimensional Banach space) with unit ball $B$. Various definitions of surface area are possible in $X$. Here we explore the one given by Benson \cite{ben1}, \cite{ben2}. In particular, we show that this definition is convex and give details about the nature of the solution to the isoperimetric problem.