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Let $X$ be a negatively curved symmetric space. In other words, $X$ is one of the four examples: a hyperbolic space, a complex hyperbolic space, a quaternionic hyperbolic space or the hyperbolic Cayle …

In addition to Anton Petrunin's answer, here is a trick to simplify (and in some sense solve) the geodesic equation. Since the metric has three-dimensional group of isometries (generated by rigid moti …

The answers are no to Question 1, and yes to Question 2 (assuming $n\ge 2$). Let $h$ be the support function of $K$. The projection of $K$ to a linear subspace $L$ is central symmetric iff the restri …

A (Euclidean) polyhedral space is a metric space obtained by "gluing together" several (let's assume finitely many) Euclidean simplices (of varying dimensions) by identifying some faces via isometries …

I am one of the authors of the reference in question. Perhaps there is a confusion between "geodesics" and "minimizing geodesics". A minimizing geodesic between $p$ and $q$ is unique, as Martin Kell e …

We need to check $\eta(u,v)+\eta(v,w)\ge\eta(u,w)$. Introduce coordinates $x,y,z$ so that the form is $x^2+y^2-z^2$. First, verify that there is a Lorentz map sending $v$ to $(0,0,1)$. Since it is an …

Here is a non-constant counter-example to the original version (with weak monotonicity). First observe that $f(\theta) $ (up to a factor $\sqrt2$ or so) equals the maximum area of intersection of $K$ …

The square case was posed as a problem at Leningrad (now St. Petersburg) high school math olympiad in 1963. I wrote a solution of this problem for the volume "St. Petersburg mathematical olympiads 196 …

Let $X$ be a finite-dimensional Banach space whose isometry group acts transitively on the set of lines (or, equivalently, on the unit sphere: for every two unit-norm vectors $x,y\in X$ there exist a …

In dimension 1, Radon's theorem says that for any 3 points on the real line, one of them belongs to the segment between the two others. This becomes false if one replaces the real line by the tripod ( …

I recently discovered the following fact: Let $K\subset\mathbb R^3$ be an origin-symmetric convex body with smooth and strictly convex boundary. Suppose that all central cross-sections of $K$ (that is …

If you allow an additive term $C(n)diam(g)$ rather than $2diam(g)$, then yes, the statement is true. In the paper D.Burago, "Periodic metrics", Adv. Soviet Math. 9, (1992), 205-210, he proves that for …

The uniform distribution on the circle is optimal. Every probability measure on the disc can be approximated by the sum of atomic measures with equal wieghts, that is, by measures of the form $\frac1 …

This is an expansion of Anton Petrunin's comment. Let me describe how to perturb the standard metric of the plane so that the resulting metric is bi-Lipschitz to the original with Lipschitz constant …

No. Let $X$ be a tripod (three segments with one common endpoint), $d=1$, $r=2$ and $E$ the set of the 3 leaf points.