There is a well known inequality due to Gage which asserts the following. Let $\Omega$ be a smooth, convex set in $\mathbb{R}^2$ and let $p = \langle X, \nu \rangle$ be the support function of $\Omega$, where $X = \langle x, y \rangle$ with respect to some origin $O$ and $\nu$ is the normal to the boundary.

Denoting $A$ and $L$ and the area and length of the curve, then it holds that $\int_{\partial \Omega} p^2 dS \leq \frac{AL}{\pi}$ for some particular choice of origin $O \in \Omega$.

**Question/Conjecture:** Given an arbitrary simply connected, smooth set $\Omega$, does it hold that $\int_{\partial \Omega} p^2 dS \leq \frac{LA^*}{\pi}$ where $A^*$ denotes the area of the convex hull of $\Omega$ and $L$ is the length of the original boundary $\partial \Omega$.

**Update June 04/2012:** There has been an answer to my original question so I would like to ask if a related although weaker assertion is true. Let $\partial \Omega$ be paramaterizable by the angle $\theta$ in polar coordinates, so that the curve is represented by $(r(\theta),\theta)$. Then $p = p(\theta)$ is obviously single valued. This means precisely that the domain $\Omega$ is *star shaped*. Let $p^*$ be the support function of the convex hull $\Omega^*$. Does it hold that $\int_{\partial \Omega} p^2 \leq \int_{\partial \Omega^*} (p^*)^2 dS$?

Any direction to references on related questions would also be greatly appreciated.

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