Lemma 8.6.2 — some endpoint lies in at least n/2d of n pairwise intersecting d-intervals

Let $d \ge 1$ and $n \ge 1$, and let $J_1, J_2, \dots, J_n$ be $d$-intervals (a finite sequence, so repetitions are allowed) such that $J_i \cap J_j \ne \emptyset$ for all $i, j \in \{1,\dots,n\}$. Then there is an index $i$ and an endpoint $p$ of $J_i$ such that

This is the counting step behind the section's capstone: in a pairwise intersecting sequence, a single endpoint is covered by a fixed fraction $1/2d$ of the members. Allowing repetitions is what later lets rational weights be replaced by multiplicities.

Formalization Note The sequence is indexed by Fin n (the book's $1,\dots,n$ are 0, …, n-1). The count is the number of indices $j$, counted with repetitions, and $n/2d$ is real division. The hypothesis $n \ge 1$ is implicit in the book's $J_1,\dots,J_n$ and is stated explicitly, since with $n = 0$ there is no endpoint at all.