**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*.