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.
