Local minimum from directional derivatives in the space of convex bodies - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T11:00:07Z http://mathoverflow.net/feeds/question/86653 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/86653/local-minimum-from-directional-derivatives-in-the-space-of-convex-bodies Local minimum from directional derivatives in the space of convex bodies Yoav Kallus 2012-01-25T20:42:20Z 2012-01-25T21:54:54Z <p>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 $d(K,B)&lt;\epsilon$ 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?</p> <p>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, $f_2(K)&lt;1$. 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.</p> <p>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?</p>