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I don't follow quite the argument in the reference but maybe the following helps: On the one hand assuming strict convexity of the inner metric there can be at most one geodesic (see (1) below). On …

For $n\ge 2$ every balloon map is a composition of a scaling, a translation and a rotation/reflection. The following proof is incomplete as it relies on the fact that the boundary of a ball is mappe …

The distance $f: x \mapsto \mathsf{dist}(x,\Gamma)$ to a closed set $\Gamma$ in $\mathbb{R}^n$ is differentiable in $x \notin \Gamma$ iff the nearest point projection is unique; denote this by $x_\Gam …

A general note is that the answer depends heavily on the properties of $\mu$. First a note that in general $d_H(A,B) \not \le C \cdot W_p(\mu|_A,\mu|_B)$ for $p\in[1,\infty)$ and some $C>0$. Though …

There can be no such function even in the category of smooth functions. Here an example for functions $f:[0,1]\to [0,1]$ with $f(0)=\epsilon$ and $f(1)=1-\epsilon$: (1) Let $\operatorname{id}:x\ma …

The article by X. Liu and Sh. Deng "The antipodal sets of compact symmetric spaces" gives many examples, e.g. $\mathrm{SU}(2n)$, $\mathrm{Spin}(5)$, $\mathrm{Spin}(7)$,.... All those spaces have uniqu …