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.

. Is it the case that the maximumQ1convexvolume body inside a torus in $\mathbb{R}^3$ is the intersection with a cylinder, as shown below?

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 $r<1$.

. 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) throughQ2anysuch tubular neighborhood radius of $\le r$ and overall central-rib $C$ curvature $\le 1$?

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

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

Related: Sofa in a snaky 3D corridor.

Q1-- it would seem you might do better by replacing the cylinder with a half-space tangent to the inner equator; at any rate you'd do no worse because this half-space contains the cylinder (if I'm reading your picture right). $\endgroup$was unwarranted. $\endgroup$Q1