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This is not an answer but an extended comment to indicate that the answer would be no without the simplicity or convexity assumptions. First observe that there are curves in $\textbf{R}^2$ with exactl …

Here is an example which shows that the answer is no. Each $\Gamma_t$ is the graph of a smooth symmetric function on $[-1,1]$. For $t\in[0,1)$, $\Gamma_t$ lies above the $x$-axis except for a pair of …

I believe that the idea described by Ian Agol works, and can be elaborated on as follows. The general fact we want to establish is that the convex hull of a finite collection $X$ of points in $M$ is n …

The following paper develops the outline described by Anton Petrunin to solve Gromov's problem on total absolute curvature for simply connected surfaces: Convexity and rigidity of hypersurfaces in Car …

Yes, closed convex surfaces in spaces of constant curvature are rigid. See Chapter V of Pogorelov's book Extrinsic Geometry of Convex Surfaces. As described at the beginning of the chapter on p. 270-2 …

In another paper with James Wenk, we have shown that the condition in Zalgaller's conjecture that the curve lie outside the sphere is not necessary, that is, the inequality $L\geq 4\pi$ holds for all …

For $n=3$ the answer is yes, as was shown by Fejes Tóth in 1943; see the Theorem on p.34 of his book Regular Figures. For $n=4$ the answer is also positive as shown in the 2000 paper, The blocking num …

In any Hadamard space, projection into convex sets is non-expansive; see Proposition 2.4(4) in Metric spaces of non-positive curvature by Bridson and Haefliger.

The answer is yes. To show this one can use the fact that any topological immersion (locally one-to-one continuous map) of an n-dimensional disk into a sphere of the same dimension is an embedding (gl …

James Wenk and I just finished a paper proving Zalgaller's sphere inspection conjecture for closed curves: Shortest closed curve to inspect a sphere. We show that in $R^3$ any closed curve $\gamma$ w …

Here is another answer to the convexity (or more precisely concavity) part of the question, which I think is even more simple than the one Anton gave: Let $p\in\Omega$, and $q$ be one of its closest …

Answer to question 1: If a convex surface is not closed, then generally it is far from rigid as it might admit infinitely many isometric deformations; however, if the surface has $\mathcal{C}^{2,1}$ r …

The answer to Question 1 is yes, which is precisely Lemma 3.6 in the paper: Boundary torsion and convex caps of locally convex surfaces, J. Differential Geom., 105 (2017), 427-486. Although the lem …

Natural higher genus analogues of convex surfaces are usually considered to be surfaces which satisfy the "two piece property" or are "tight". A closed surface in Euclidean space is said to have the …

I have recently finished a paper called The length, width, and inradius of space curves where it is shown that the length $L$ of any closed curve $\gamma\colon[a,b]\to \mathbf{R}^3$ inspecting the u …