Kneser–Poulsen Conjecture

Let \(d,N\geq1\), \(r>0\), and
\(p_1,\ldots,p_N,q_1,\ldots,q_N\in\mathbb R^d\) satisfy
\[
  \lVert q_i-q_j\rVert_2\leq\lVert p_i-p_j\rVert_2
  \quad\text{for all }i,j.
\]
Writing \(B(x,r)\) for the closed Euclidean ball and
\(\operatorname{vol}_d\) for \(d\)-dimensional Lebesgue measure, one has
\[
  \operatorname{vol}_d\!\left(\bigcup_{i=1}^N B(q_i,r)\right)
  \leq
  \operatorname{vol}_d\!\left(\bigcup_{i=1}^N B(p_i,r)\right)
\]
and
\[
  \operatorname{vol}_d\!\left(\bigcap_{i=1}^N B(q_i,r)\right)
  \geq
  \operatorname{vol}_d\!\left(\bigcap_{i=1}^N B(p_i,r)\right).
\]