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$\newcommand{\R}{\mathbb R} \newcommand{\dd}{\operatorname{d}\!}$ This is not a complete answer, but it may lead to finding the asymptotics of the surface area; an exact closed-form expression for th …

$\newcommand\R{\mathbb R}$This is not true in general. E.g., let $p(s):=(\arctan s,0,\dots,0)$. Then $$\bigcup_{s\in\R}N_s=(-\pi/2,\pi/2)\times\R^{n-1}\ne\R^n.$$

By the fundamental theorem of geometry, if a transformation of $\mathbb R^d$ with $d\ge2$ is bijective and maps lines onto lines, then it is affine. The main result in AMS Proc. 1999 implies that, mor …

Let $d(x,Y):=\min\{\|x-y\|\colon y\in Y\}$ and $d(y,X):=\min\{\|y-x\|\colon x\in X\}$. Then $$d(X,Y)=(1-w)d_X(Y)+wd_Y(X),$$ where $$w:=\frac{|Y|}{|X|+|Y|},$$ $|X|$ and $|Y|$ are the cardinalities of $ …

As Francois Ziegler pointed out, the result that you need is the so-called fundamental theorem of geometry (FTG), which states the following: if a transformation of $\mathbb R^d$ with $d\ge2$ is bijec …

$\newcommand\C{\mathscr C}\newcommand\ep{\varepsilon}$Let $\C$ denote the set of all disks of a radius $r\in(0,\infty)$ contained in the unit square. Using a rectangular grid of centers of disks in $\ …

Yes, this follows immediately from the triangle inequality on the sphere -- cf. e.g. this or this.

The case $d=1$ is simple. Then the probability in question is $$p:=P(V-U\le r),$$ where $U$ and $V$ are, respectively, the smallest and largest order statistics for an iid random sample of size $n$ fr …

$\newcommand{\R}{\mathbb R} \newcommand{\dd}{\operatorname{d}\!}$ This is not a complete answer, but it may lead to one. Since the volume of the unit $\ell_p$ ball $B_p^d$ is known in closed form, we …

Let us change notation somewhat: Let $x=(x_1,\dots,x_n)$ and $y=(y_1,\dots,y_n)$ be points in $\mathbb R^n$ such that $|x|=|y|=1$ and $|x-y|=d\in(0,1)$, where $|\cdot|$ is the Euclidean norm. Let $v$ …

$\newcommand\tX{\tilde X}\newcommand\tY{\tilde Y}\newcommand\D{\overset D=}$The answer is no. E.g., suppose that $n=2$, so that $\tX_1,\tX_2,\tY_1,\tY_2$ can be identified with $X_2,X_1,Y_2,Y_1$, res …

$\newcommand\De\Delta\newcommand\de\delta$Yes, this map is Lipschitz. Indeed, the map is \begin{equation*} \De\ni w=(w_1,\dots,w_n)\mapsto\mu_w:=\sum_{k=1}^n w_k\de_{x_k}. \tag{1} \end{equation*} …

Yes, you do need more conditions. For instance, if $f(x_1,\dots,x_n)\equiv x_1+\dots+x_n$ and the $X_i$'s are (say) iid Bernoulli with parameter $1/2$, then $f$ is $1$-Lipschitz in the $L^1$-norm but, …

$\newcommand\om\omega\newcommand\R{\mathbb R}$In general, even the inequality $$\om_g\le\om\circ\om_f^{-1}\tag{0}$$ will not hold, for the right inverse $\om_f^{-1}$ of $\om_f$ defined by $$\om_f^{-1} …

$\newcommand\ep\varepsilon\newcommand\R{\mathbb R}$Suppose that $G$ is a measurable subset of $\R^N$ with volume $|G|>0$ such that $$|G|\le C^N|B|,$$ where $C>0$ is a real constant and $B$ is the unit …