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?

  • $\begingroup$ Blaschke compactness theorem alone will not help, you need to look at more properties of $f$. Your function $f_2$ is a generalized width and you want the ratio $\sqrt[3]{vol}/f_2$ to have minimum for ball --- for the usual width it is not true and I guess that for most of "generalized widths" it should not be true. (My intuition says: "it should be true only for completely symmetric $\mu$".) $\endgroup$ – Anton Petrunin Jan 25 '12 at 23:18
  • $\begingroup$ It is true for the minimum width, of course. Can you explain what you mean by "generalized widths" and also by "completely symmetric $\mu$"? The minimum width corresponds to the above $f_2$ with a measure supported at a pair of antipodal points. $\endgroup$ – Yoav Kallus Jan 26 '12 at 1:13
  • $\begingroup$ generalized widths is your $f_2$, say if support of $\mu$ is formed by two opposite points of $S^2$ then $f_2$ is the standard width of $K$. $$ $$ Completely symmetric means invariant w.r.t. all rotations. (I.e. $\mu$ is proportional to Lebesgue measure.) $\endgroup$ – Anton Petrunin Jan 26 '12 at 4:24
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    $\begingroup$ Actually, for symmetric $\mu$ (i.e. $w=$mean width), $\sqrt[3]{vol}/w$ is (globally) maximized by balls (aka Urysohn's inequality). For the standard width, the ball is locally neither maximal (see e.g. ellipsoids) nor minimal (e.g. bodies of constant width). What makes my $\mu$ different is the condition that the projection to the space of spherical harmonics of degree $n$ vanishes only for $n=1$. This is certainly untrue for the mean width (vanishes for all $n>0$) or for the standard width (vanishes for odd $n$). If it helps, my $\mu$ is supported at $12$ points with equal weight at each. $\endgroup$ – Yoav Kallus Jan 26 '12 at 6:15
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    $\begingroup$ P.S. It seems that for my generalized width, the ratio is smaller for the regular tetrahedron than for the ball, so the ball is not a global minimum (as it is for standard width amongst c.s. bodies), but I still think it's a local minimum. $\endgroup$ – Yoav Kallus Jan 26 '12 at 21:06

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