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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 …

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 …

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 …

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.

This is not true even in finite dimensions. There exists a decreasing sequence of complete Riemannian metrics on the plane, pairwise Lipschitz equivalent, such that the pointwise limit is isometric to …

The optimal shape is the regular $n$-gon and all its affine images. I am going to optimize the ratio $$ \frac{\sum_{i<j} |P_iP_j|^2}{\sum_i|P_iP_{i+1}|^2} $$ which differs from $A_n$ by 1. For the r …

Dealing with such a low regularity is a tricky business. However in dimension 3 you can get away with your set of assumptions. First, if the distributions are uniformly Lipschitz and converge in $C^0 …

For the Lipschitz-only part, the answer is yes. More generally, if $\alpha$ and $\beta$ are any two convex curves and $\beta$ fits inside an $L$-homothetic copy of $\alpha$, then there exist an $L$-Li …

There is no Lipschitz or even Gromov-Hausdorff stability - just consider a round metric with long hairy tails of small area. One can hope for stability with respect to intrinsic flat distance in the …

$3^d-1$ and $3^d$, resp. Let $C=[0,1]^d$, Consider the $3^d$ points in $C$ all whose coordinates are from the set $\{0,\frac12,1\}$. No translated copy of $C'$ can cover two of these points, hence at …

Yes it is true. Let me show that the existence of $\phi$ implies that the norm of $B^*$ is associated to an inner product. Then it follows easily that the spaces are Hilbert. It suffices to verify th …

No, the Loewner ellipsoid is not monotone w.r.t. inclusion. Let $K$ be a square, whose Loewner ellipsoid is its circumcircle. Let $L$ be any other ellipse through the four vertices of $K$. The Loewner …