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Let $\mathcal K^3$ be the space of convex bodies, with some metric $\delta$, and let $B$ be the unit ball. Let us define a volume-preserving perturbation of the sphere to be a continuous (substitute Lipschitz or something else if necessary for the sequel) map $K:[0,\epsilon_0]\to \mathcal K^3$ where $\epsilon_0>0$, $\delta(K(\epsilon),B) = \epsilon$, and $V(K(\epsilon))=V(B)$. Let $\delta(K,B) = V(K\setminus B)+V(B\setminus K)$, which is the symmetric difference metric, and let $\rho_K(x) = \max_{\lambda x\in K} \lambda$, which is the radial distance function, then I have functions $f_\epsilon(x) = (\rho_{K(\epsilon)}^3(x)-1)/3\epsilon$ that for $\epsilon>0$ all have $\int_{S^2}|f_\epsilon(x)|d^2x=1$ and $\int_{S^2}f_\epsilon(x)d^2x =0$.

Q1. Under what reasonable conditions can I say that $\lim_{\epsilon\to0}f_\epsilon = f_0$ where $f_0(x)$ is a signed measure? What kind of convergence will I have?

Q2. The signed measure $f_0(x)$ is in a sense the derivative $d\rho_{K(\epsilon)}(x)/d\epsilon$ at $\epsilon=0$. Can this relationship between the limit measure and the point-wise derivative be made exact by saying for example that $\lim\inf (\rho_{K(\epsilon)}(x)-1)/\epsilon = f_0(x)$ (i.e. the point-wise Dini derivative is given by the density of the limit measure)?

Q3. I would much rather have a result in terms of $h_K(x)$ (the support height) rather than $\rho_K(x)$ (the radial distance). Can a statement such as in Q2 also be made for the point-wise derivative of $h_{K(\epsilon)}(x)$? For example if I let $\delta(K,B)=\int_{S^2}|h_K(x)-1|d^2x$ and let $f_\epsilon = (h_{K(\epsilon)}(x)-1)/\epsilon$? Will I still have $\int_{S^2} f_0(x) d^2x = 0$?

Edit: I did a bit of reading and figured out part of my question. As I understand, Prokhorov's theorem guarantees that $f_\epsilon$ have a weak limit $f_0$. Also, some simple examples such as the convex hull of a sphere and a point above the north pole show that no stronger convergence is possible. So if I understood my reading correctly, that settles Q1. Q2 boils down to whether the limit taken in Q1 and the limit involved in taking the Radon-Nikodym derivative can be interchanged. I have not been able to find information and when that is possible.

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What is $\rho$? –  Anton Petrunin Apr 19 '12 at 18:34
    
Sorry. $\rho_K(x)$ is what I call the radial distance function. I.e. $K = \{\lambda x\rho_K(x) : x\in S^2, 0\le\lambda\le 1\}$. Edited the question to make that more clear. –  Yoav Kallus Apr 19 '12 at 20:46

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