Search Results

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 …

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 …

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 …

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 …

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 …

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 …

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 …