I have a function $f(K)$ defined on the space of three-dimensional convex bodies for which I want to show that the unit ball $B$ is a local minimum. I have been able to show if $K$ is not homothetic to $B$, then the function $g(\alpha) = f(K_\alpha) = f((1-\alpha)B+\alpha K)$ is positively sloped at $\alpha=0$. (If it is homothetic then $g(\alpha)$ is constant). This, of course, implies that for any $K$, there is $\alpha>0$ such that $f(K_\alpha)\ge f(B)$. However, what I actually want to show is that there exists $\epsilon$ such that if $d(K,B)<\epsilon$ then $f(K)\ge f(B)$ (where $d$ is the Hausdorff metric). Question: can I use the fact that the space of convex bodies is locally compact (i.e. the Blaschke Selection Theorem) to go from one result to the other?
Here are some more background and details which might be useful: the function is defined as a ratio $f(K)=f_1(K)/f_2(K)$ where $f_1(K) = V(K)^{1/3}$ is the cube-root of the volume, and $f_2(K) = \min_{\vartheta\in SO(3)} f_\vartheta (K)$ is the minimum of a family of functions that are each linear in the support height function of $K$, $h_K(\mathbf{u})$. Namely, $f_\vartheta(K) = \int_{S^2} h_K(\vartheta(\mathbf{u})) d\mu(\mathbf{u})$, where $\mathbf{u}\in S^2$, $\vartheta\in SO(3)$, and $\mu$ is some measure on $S^2$ that has $\mu(S^2)=1$ and $\int_{S^2} \mathbf{u} d\mu(\mathbf{u}) = 0$. Therefore, $f(\lambda K + \mathbf{t}) = f(K)$, and we can limit our attention to bodies $K$ with a mean width of $2$ and Steiner point at the origin. I have that the projection of $\mu$ to the space of spherical harmonics of degree $n$ never vanishes for $n>1$, and therefore $f_2(K) = 1$ if and only if $K=B$; otherwise, $f_2(K)<1$. Since $g_1(\alpha) = f_1(K_\alpha)$ has zero slope at $\alpha=0$ (by the definition of mixed volumes, the slope is given by the difference in mean widths of $K$ and $B$) and $g_2(\alpha) = (1-\alpha) + \alpha f_2(K)$, then $g(\alpha)$ is positively sloped at $\alpha=0$. I have tried to put more definite bounds on $f_1$ and $f_2$ as a function of $h_K(\mathbf{u})$. I think I can obtain $f_1(K)-f_1(B) \ge -c ||\nabla_0 h_K||^2$ (i.e. the $L^2$ norm of the magnitude of the gradient of the height function restricted to the sphere) and $f_2(K)-f_2(B)\le c' (\min_\mathbf{u} h_K(\mathbf{u}) - 1)$ (but not, it seems, $-c'' (\max h -1)$), but I'm not sure those give me anything.
If the answer to my original question is no, can you suggest a way to obtain my desired result ($B$ is a local minimum of $f$) by other means?
