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This question is inspired by the discussion in MO questions "Local minimum from directional derivatives in the space of convex bodies" and "Bodies of constant width?" about generalized notions of minimum widths and constant widths. Some interesting ideas came up in those discussions which I would like to bring together with some ideas I had and some things I'm not clear on yet.

Let the $\mu$-width of a three-dimensional body $K$ be a function $$w_\mu(\vartheta)=\int_{S^2} h(\vartheta(\mathbf{u})) d\mu(\mathbf{u})\text,$$ where $\vartheta\in SO(3)$ is a rotation, $h(\mathbf{u})=\max_{\mathbf{x}\in K} \mathbf{u}\cdot\mathbf{x}$ is the support height function, and $\mu$ is a (signed?) measure on $S^2$ with $\int \mathbf{u} d\mu(\mathbf{u})=0$ (to ensure translation invariance). The standard width corresponds to a measure concentrated at two opposite points. The mean width corresponds to the uniform measure. The main properties I want to talk about are the minimum $\mu$-width (the minimum of $w_\mu(\vartheta)$) and bodies of constant $\mu$-width. The mean $\mu$-width is not interesting because it reduces to the mean standard width (assuming $\mu(S^2)\neq 0$).

The minimum standard width is the "mailslot width", the smallest mailslot through which the body can pass. If $\mu$ is concentrated uniformly on the equator (this is related to the spherical Radon transform) then the minimum width is the "loop width", giving the smallest length of string loop through which the body can pass (this follows from the fact that the mean width of a planar body is proportional to its perimeter). If $\mu$ is concentrated with equal weight at the vertices of a regular tetrahedron, the minimum "tetrahedral width" gives the linear size of the smallest regular tetrahedron that contains the body. Bodies of constant tetrahdral width are the rotors of a tetrahedral cavity (see "Bodies of constant width?").

Let us define the "harmonic support" (h.s.) of a function or measure on $S^2$ as the set of integers $n>1$ such that the projection of the function to the space of spherical harmonics of degree $n$ does not vanish. Also let $\mathcal{K}_I$ be the space of convex bodies such that the harmonic support of their height function is a subset of $I$. If $I$ and $J$ are disjoint and their union is $\{n>1\}$, I call $\mathcal K_I$ and $\mathcal K_J$ complementary spaces. Then the space of bodies of constant $\mu$-width is the complementary space to $\mathcal{K}_{h.s.(\mu)}$. Thus, it follows that bodies of constant width and bodies of constant loop width are the same.

Urysohn's inequality says that the ratio $vol/\bar{w}^3$, where $\bar{w}$ is the mean width is maximized by balls. In general, the ball also maximizes the ratio $vol/w_\mu^3$ among bodies of constant $\mu$-width $w_\mu$. I am interested in the complementary space, and whether $vol/w_\mu^3$, where $w_\mu$ is the minimum $\mu$-width, is minimized by balls among bodies in $\mathcal K_{h.s.(\mu)}$. Clearly, this holds for the standard width: among all centrally-symmetric bodies of a given volume, balls maximize the mailslot width (not true if central symmetry is not assumed). However, based on some experiments I made, I find that this is not true in general as a global statement. Still, I believe that balls are local minima. This is because it is pretty easy to show that if $K\in\mathcal K_{h.s.(\mu)}$ is not a ball, then for some $\alpha_0>0$ the body $K_\alpha=(1-\alpha)B+\alpha K$ obtains a greater ratio than that of the ball for all $0<\alpha<\alpha_0$. (See "Local minimum from directional derivatives in the space of convex bodies"). My question is, can you find a counterexample of my claim that $B$ is a local minimum of $vol/w_\mu^3$ among bodies in $\mathcal K_{h.s.(\mu)}$; or can you see a way of proving it?

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Dear Yoav, Do you have some idea on how $\alpha_0$ depends on $K$? I guess you need some sort of uniform estimate. Is $K=B$ the only "problem"? By this I mean that as $K$ approaches a ball the estimate of $\alpha_0$ will get worse and worse because $f(K_\alpha)$ will be constant as a function of $\alpha$, but is the ball the only convex body for which you see this happening? I struggled with a problem like this and at the end settled for the directional derivative result ... The problem was that there were lots and lots of directions for which the directional derivative was zero. –  alvarezpaiva Feb 12 '12 at 12:15
    
Dear Prof Alvarez, thank you for your comment. I am also leaning toward settling for the partial result, but not there yet. The directional derivative is strictly positive in my case for all $K\neq B$. In fact, because $\alpha_0(K)$ is continuous (I think), there is a positive minimum $\alpha_0(\epsilon)$ over all $K$ with $d(K,B)=\epsilon$ by compactness. However, not all $K'$ with $d(K',B)\le\epsilon$ can be written as $K_\alpha$ where $d(K,B)=\epsilon$. –  Yoav Kallus Feb 12 '12 at 20:43
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This is hardly a direct answer to your question, but a new paper—at least tangentially relevant—by HaiLin Jin and Qi Guo addresses the question of how assymetric can a constant-width body be. In "Asymmetry of Convex Bodies of Constant Width" (Discrete & Computational Geometry Vol. 47, No. 2, Mar. 2012, 415-423), they establish tight bounds on the "Minkowski measure of asymmetry for convex bodies." In particular, they extend the known result that Reuleaux triangles are most assymetric in $\mathbb{R}^2$ to showing that Meissner's tetrahedron is most assymetric in $\mathbb{R}^3$. An image of Meissner's tetrahedron appeared in the earlier MO question, "Are there smooth bodies of constant width?"

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Thanks for beautifying my OP. Though your answer doesn't directly address my question, it does suggest other questions we may ask about generalized widths. E.g., we may ask how asymmetric can a body of constant tetrahedral width be. This is much simpler then the question for constant standard width, because the corresponding space only has 13 d.o.f. (fixing dilation, translation, and rotation). Even simpler if we discuss rotors of a octahedral cavity (only 8 d.o.f.). –  Yoav Kallus Jan 27 '12 at 22:25
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