Sliding through a curvature-bounded tube: Maximum volume?

Question

My 1st question has a straightforward answer but I'd appreciate hints on a proof. My 2nd question is open from my point of view.

Q1. Is it the case that the maximum convex volume body inside a torus in $\mathbb{R}^3$ is the intersection with a cylinder, as shown below?

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Let $C$ be a smooth curve in $\mathbb{R}^3$, whose maximum curvature at any point $x \in C$ is $\le 1$. Now consider a tubular neighborhood of $C$— (used also in Light rays bouncing in twisted tubes)— width of $r<1$.

Q2. Let the curvature of the smooth $C \in \mathbb{R}^3$ be bound by $\le 1$. What is (a description of) the maximum volume convex shape that could move (via rigid motions) through any such tubular neighborhood radius of $\le r$ and overall central-rib $C$ curvature $\le 1$?

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          ^{A smooth curve $C$ with curvature everywhere $\le 1$. Tube of radius $r < 1$.}

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I presume the optimal shape is convex. I suspect this question has been considered previously...?

Related: Sofa in a snaky 3D corridor.

Answer 1

Not being a geometer, I have a hazy intuition for a proof of question 2, which hopefully someone else can formalize.

I interpret the problem as finding a solid which can pass (using rigid motions) through a constraining tube T, where constraints are that T looks like it "bends no more sharply" than a torus of similar radius and curvature. Indeed, if we have such a curve on a plane, create the associated tube, and then consider the intersection of "enough pieces" of the tube, that intersection should pass through T. Indeed, if T is made of toroidal arcs of a given curvature joined by cylinders, an intersection should look like Joseph's first shape giving intersection of a torus and a cylinder.

(Pause for a proof hint the Joseph's shape is optimal: given a circle C cutting the torus into a shape of genus 0, consider the transverse circles which indicate concavity of the torus. At most one point of the transverse circle can lie in a convex body, and so that point lies on C. Now use tangent hyper planes to C to get Joseph's shape.)

To strengthen the intuition, imagine the containing cylinder being slowly bent around Joseph's shape, resembling tori of sharper curvature, until the limit is reached. The intersection of all those tori is also Joseph's shape, or the intersection of two them, since the curve can bend to the left or to the right in the plane. (We still have our original curve in a plane.) Part of my intuition here which needs professional geometric verification is that for this case, a tubular neighborhood at a point is well approximated by joining two pieces of tori together, possibly of two different curvatures.

Now to mildly generalize out of the plane. Instead of approximating a piece of T at a point by gluing two pieces of tori together "with zero torsion", twist one of the pieces slightly to create a neighborhood for a curve that rises out of the plain. Now consider intersecting a lot of these twisted neighborhoods together. The intersection will be strictly contained in Joseph's shape. How do we show this intersection is convex?

I believe we can do it by using the tangent planes of the tube. As the shape slides through all possible tubes, this means it belongs to all possible neighborhood shapes appearing, and so to their intersection. It is my belief that the shape is actually contained in the intersection of all tori of optimal curvature that share the cutting circle C, so itself a solid of revolution. Hopefully someone who knows geometry can fill in the gaps.

Gerhard "Holes Not Just In Tori" Paseman, 2018.06.08.

Answer 2

I want to say an intuition on question 1 :

Def :

$$ f(t,s):=\bigg((R +r\cos\ t)\cos\ s,(R+r\cos\ t)\sin\ s,r\sin\ t\bigg) $$

Define a solid torus $T:={\rm conv}\ f([0,2\pi]^2)$ and $d$ to be a Euclidean distance in $\mathbb{E}^3$.

Def : Define $$S_t=f\bigg( t,[0,2\pi)\bigg) ,\ \frac{\pi}{2} < t < \frac{3\pi}{2} $$

Property of Convex Set : Assume that $C$ is a closed convex subset in $T$. Then $$ \bigg|\ S_t\bigcap C \ \bigg|\leq 1 $$

EXE : Let $p\in S_t$. Note that tangent a plane $T_p \partial T$ cuts $ T $ into two components $T_i$. Then one component $\overline{T}_1\bigcap B^d(p,\epsilon)$ is star-shaped at $p$ in $ \mathbb{E}^3$.

In further, if a plane at $p$ is close to $T_p\partial T$ s.t. it has such property, then it is the tangent plane.

Proof : By Gaussian curvature condition, any geodesic in $\partial T_1$ at $p$ has a positive curvature.

EXE : Fix $p_0\in S_\pi$. Then $T_{p_0} \partial T$ cuts $T$ into two components. Clearly, if $C$ is a closed convex set, then it can be assumed in $T_1$.

So now we suffice to rule out points in $\partial T_1$ of negative curvature. If $c(t)\in S_t,\ 0<\frac{\pi}{2}<\frac{3\pi}{2}$ is a curve, then we cut $T_1$ with a surface $M_{c,v} \ :\ x(t,s)=c(t)+sv(t)$ where $v(t)$ is a unit field in $T_{c(t)}\partial T_1$.

Here, $M_{c,v}$ is a ruled surface so that it is nonpositively curved. Hence when it is flat, the resulting body has a maximal volume.