# Is there a convex three-dimensional body with constant width and only finitely-many equilibria? Or: do spheroform gömböcök exist?

Mathematical questions. The mathematical (and 'gravity'-free) formulation of the question in the title is given by the following questions:

Q1. Does there exist $(a,b)\in\omega^2\setminus\{(0,0)\}$ such that there exists a function $R\colon\mathbb{R}\times\mathbb{R}\rightarrow \mathbb{R}$ such that

$S_R:=\{ ( {\small R(\theta,\varphi)\cdot\cos\theta\cdot\sin\varphi},{\small R(\theta,\varphi)\cdot\sin\theta\cdot\sin\varphi},{\small R(\theta,\varphi)\cdot\cos\theta})\colon {\small(\theta,\varphi)\in\mathbb{R}\times\mathbb{R}}\}\subseteq\mathbb{R}^3$

satisfies

(1) ${}\qquad$ $S_R$ is a convex surface

and

(2) ${}\qquad$ $S_R$ is a surface of constant width(1)

and

(3) ${}\qquad$ there is $(a,b)\in\omega^2$ such that the scalar field $\mathbb{R}^2\xrightarrow[]{(\theta,\varphi)\mapsto R(\theta,\varphi)}\mathbb{R}$ has exactly $a$ stable and exactly $b$ unstable critical points

?

Q2. What is the answer to Q1 with "Does there exist $(a,b)\in\omega^2\setminus\{(0,0)\}$ such that there exists a function $R\colon\mathbb{R}\times\mathbb{R}\rightarrow \mathbb{R}$ such that" replaced by "Is it true that for all $(a,b)\in\omega^2$ there exists a function $R\colon\mathbb{R}\times\mathbb{R}\rightarrow \mathbb{R}$ such that" ?

Bibliographic question. Has this question been raised anywhere in the literature?

Remarks.

• I find it hard to believe that it hasn't been asked anywhere, and yet, despite searching much, I didn't find anything. This is hard to believe, especially in view of the attention that gömböcök garnered over the last decade; I'd expect that anyone who has seen some gömböcök and also seen some body of constant width, would ask themselves whether there's a body being both, wouldn't they?

• Needless to say, the physical interpretation of the above is that $R$ represents a perfectly homogeneous and perfectly inelastic body with non-zero mass which in a perfectly parallel and homogeneous field of gravity orthogonal to a perfectly inelastic and perfectly plane infinite surface has exactly $a$ stable and exactly $b$ unstable equilibrium points.

• Needless to say, for $S_R$ to be a convex surface, $R$ must be periodic in both the first and the second argument. Spelling this out though, or using one of the finite intervals customary when using spherical coordinates, seems irrelevant.

• Since there are infinitely-many non-isomorphic surfaces of constant width, and also infinitely-many non-isomorphic gömböcök, there seems to be some hope that the intersection of the two classes of convex surfaces is non-empty.

• An $R$ satisfying (1)--(3) with $(a,b)=(1,1)$ would amount to a gömböc of constant width. (No one seems to asked for this before; is there an easy proof that such gömböcök are impossible?)

• I don't know why, but I am most interested in the simplest case $(a,b)=(1,0)$, which amounts to asking for a perfectly homogeneous and inelastic roly-poly toy of constant width. (I suspect there is an easy proof of its impossibilty, but can't seem to find one.)

• None of the gömböcök I have 'seen' seem to have constant width. Based upon a cursory reading of the work of Domokos and Várkonyi I think I am half-sure that one can use their methods to push the ratio

$\mathrm{roundedness}(R) := \frac{\inf\{\text{distances of parallel supporting planes of$S_R$}\}}{\sup\{\text{distances of parallel supporting planes of$S_R$}\}}$

arbitrarily close to $1$, but

• only at the expense of increasing $a$ and $b$, and
• $\mathrm{roundedness}(R)$ cannot be made exactly equal to $1$ via the methods of Domokos and Várkonyi,

so that for an affirmative answer to Q1 new methods seem to be required. (And I still suspect that there is a short proof that the answer to Q1 is negative.)

${}$_________________________

(1) Hence, necessarily, $S_R$ is also a surface of constant girth.