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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 …

Let $M$ be a compact $3$-dimensional manifold diffeomorphic to a ball. Suppose that $M$ has nonpositive (sectional) curvature and its boundary $\partial M$ is convex, or even that $M$ is a Riemannian …

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 …

Let $pxy$ be a triangle in a CAT(0) space $X$, and $p' x' y'$ be a triangle in $\mathbf{R}^2$ such that the lengths $|px|=|p'x'|$, $|py|=|p'y'|$ and the angle $\angle(xpy)=\angle(x'p'y')$. Let $z\in x …

This is just a bit more elaboration on Anton's nice example, and subsequent comment. Here is the picture of the two triangles: Note that $p'x'y'$ is obtained from $pxy$ by rotating the side $\overlin …

There is a problem in Richard Kenyon's list (Wayback Machine) which I would like to post here, because although I have thought about it from time to time, I have not been able to make the slightest pr …

Can the convex hull of $3$ points in a Cartan-Hadamard manifold be smooth? A Cartan-Hadamard manifold $M$ is a complete simply connected manifold with nonpositive curvature (so it includes the Euclide …

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 …

How can one construct examples of Riemannian manifolds which have nonpositive radial curvature about some point, but are not nonpositively curved everywhere? (I presume that they exist, but do not kn …

Let $X$ be a convex set in Euclidean space $\mathbf{R}^n$ and $p\in\mathbf{R}^n$ be a fixed point. Then any plane $\Pi$ passing through $p$ intersects $X$ in a convex set. Conversely, this property q …

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 …

A Cartan-Hadamard 3-space $M$ is a complete simply connected 3-dimensional Riemannian manifold with nonpositive sectional curvature. A (smooth) convex surface $\Gamma\subset M$ is an embedded topologi …

In a recent paper, Centers of disks in Riemannian manifolds, Igor Belegradek and I study whether it is possible to extend to nonconvex objects the notion of center of mass or other classical centers a …

Igor Belegradek and I just finished another paper where we construct a continuous point selection from the interior of Jordan domains in Riemannian surfaces, which is equivariant under isometries of t …

Consider the space $J$ of Jordan domains on the sphere $\textbf{S}^2$, i.e., continuous injective maps from the unit disk into $\textbf{S}^2$ modulo homeomorphisms of the disk. How can one construct a …