In this note, Gromov proved the following claim:
Let $k \le n+1$. Let $x_i \in \mathbb{R}^n$, $i=1,....,k$, and fix some radii $r_i \ge 0$. Denote by $B(x_i,r_i)$ the closed ball in $\mathbb{R}^n$ around $x_i$ with radius $r_i$. Define $ V(x_i,r_i)= \operatorname{Vol} (\cap_{i=1}^k B(x_i,r_i))$.
Then, $V$ is monotone decreasing in $d_{ij}=\|x_i-x_j\|$ (if we decrease the pairwise distance between the centers of the balls, while keeping the radii fixed, the volume of intersection increases).
Does the claim hold for $k > n+1$?
I will enumerate some known results and conjectures
Kirszbraun : In $\mathbb{R}^m$, $$ |a_ia_j|\geq |b_ib_j| $$ Then $$ \bigcap_{i=1}^n B_R(a_i)\neq \emptyset\Rightarrow \bigcap_{i=1}^n B_R(b_i)\neq \emptyset $$
Def : In $\mathbb{R}^m$ with $m>2$, ${\bf a}=(a_1,\cdots ,a_N)$ is expansion of ${\bf b}= (b_1,\cdots, b_N)$ if $ a_i,\ b_i\in \mathbb{R}^m$ and $$\ast\ |a_ia_j|\geq |b_ib_j|,\ 1\leq i,\ j\leq N $$
If $ {\bf a}$ is expansion of ${\bf b}$, then the followings are conjectured :
Conj 1 - [Poulsen 54 and Kneser 55] $$ {\rm vol}_m\ \bigcup_{i=1}^N B_{R_i}(a_i) \geq {\rm vol}_m\ \bigcup_{i=1}^N B_{R_i}(b_i) $$
Conj 2 - [ Klee and Wagon ] $$ {\rm vol}_m\ \bigcap_{i=1}^N B_{R_i}(a_i) \leq {\rm vol}_m\ \bigcap_{i=1}^N B_{R_i}(b_i) $$
Bezdek and Connelly 02 proved Conj 1 and Conj 2 for $m=2$ and $R_1=\cdots =R_N$, by using the followings :
Alexander : Let ${\bf a}_i,\ {\bf b}_i\in \mathbb{R}^n$. Define $$ \alpha_i(t) = \bigg(\frac{\textbf{a}_i+\textbf{b}_i}{2} +\cos\ (\pi\cdot t) \cdot \frac{\textbf{a}_i-\textbf{b}_i}{2}, \sin\ (\pi\cdot t) \cdot \frac{\textbf{a}_i-\textbf{b}_i}{2} \bigg) \in \mathbb{E}^{2n} $$
Then $l(t)=|\alpha_i(t)-\alpha_j(t)|$ is monotonic
Archimedes' Theorem : If $\Pi : S^2\rightarrow \mathbb{R}$ is coordinate projection then $$ {\rm area}\ \Pi^{-1} ([a,b])=2\cdot \pi\cdot (b-a) $$
EXE : Also Bezdek and Connelly introduce the exercise : In $\mathbb{R}^2$, if ${\bf a}$ is expansion of ${\bf b}$, then $\bigcup_i \ B(b_i,\epsilon)$ has larger perimeter than that of $\bigcup_i\ B(a_i,\epsilon)$
Gromov and Capoyleas and Pach : They solved Conj 1 and 2 for $N\leq m+1$ :
In these case expansion is continuous expansion. However in case of $N=m+2$ expansion does not imply continuous. As far as I know, $N>m+1$ case is not solved.