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Source. Published p. 232, Lemma 3.12, where it is quoted from Frankl–Rödl (1990). The finite cut-vector proof below is supplied by the compilation. (canonical PDF).

As printed (p. 232): for every integer $d\ge1$ there is a real $1\ge\mu=\mu(d+1)>0$ such that, for every $(\mu,\beta)$-regular simplex $T=\{t_1,\ldots,t_{d+1}\}$, there is a $\binom{d+1}2$-dimensional box $P$ (the vertex set of a rectangular parallelepiped) containing a subset $T'$ congruent to $T$.

Array form proved here. Let $n=d+1\ge2$, $M=\binom n2$, and choose

$$\mu_n=\frac1{n2^n}.$$

For every $\beta>0$, every symmetric zero-diagonal array $e$ with

$$|e_{ij}/\beta-1|\le\mu_n\quad(i\ne j)$$

is realized by an affinely independent set of $n$ vertices of a box $P$ with at most $M$ nonconstant coordinates. The box can be chosen so that

$$\rho(P)^2\le\frac{M\beta(1+\mu_n)}4 \le\frac{M\beta}{2}<\beta n^2.$$

Thus the source's box dimension can be read as an upper bound, with unused constant coordinates omitted. The array need not be assumed Euclidean in advance.

Proof.

Index the coordinates of $\mathbb R^M$ by unordered pairs of $[n]$. For every nonempty proper subset $S\subset[n]$, let $\delta_S$ have coordinate one on pairs separated by $S$ and zero on other pairs. Also write $\delta_{[n]}=0$. Each pair is separated by exactly $2^{n-1}$ subsets, so

$$\sum_{\varnothing\ne S\subsetneq[n]}2^{1-n}\delta_S=\mathbf1.$$

If $E_{ij}$ is the unit vector for pair $\{i,j\}$, then

$$2E_{ij}=\delta_{\{i\}}+\delta_{\{j\}}-\delta_{\{i,j\}}.$$

This identity also holds when $n=2$, since the final cut is zero. Put $h_{ij}=e_{ij}/\beta-1$. Start with coefficient $w_S=2^{1-n}$ on every nontrivial cut and, for each pair $ij$, add $h_{ij}/2$ to the two singleton-cut coefficients and subtract $h_{ij}/2$ from the $\{i,j\}$ coefficient if that cut is nontrivial. Then

$$\sum_S w_S\delta_S=e/\beta.$$

For $n\ge3$, a singleton coefficient changes in absolute value by at most $(n-1)\mu_n/2$, a two-element coefficient by at most $\mu_n/2$, and other coefficients do not change. For $n=2$, each singleton changes by at most $\mu_n/2$. These bounds are all strictly smaller than $2^{1-n}$, so every $w_S$ is positive.

If more than $M$ cuts have positive coefficients, they are linearly dependent in $\mathbb R^M$. Choose a nonzero relation $\sum_S\lambda_S\delta_S=0$, with some $\lambda_S>0$ after changing its sign if needed, and set

$$t=\min_{\lambda_S>0}\frac{w_S}{\lambda_S},\qquad w'_S=w_S-t\lambda_S.$$

All new coefficients are nonnegative, at least one is zero, and the represented array is unchanged. Repeating this finite operation leaves at most $M$ positive coefficients.

For each surviving cut $S$, give the box a coordinate edge of length $\sqrt{\beta w_S}$, and give its $i$-th selected vertex that coordinate exactly when $i\in S$. The squared distance between vertices $i,j$ is $\beta\sum_Sw_S\delta_S(i,j)=e_{ij}$. The points are distinct because $e_{ij}\ge\beta(1-\mu_n)>0$.

They are also affinely independent. For a zero-sum vector $\lambda$ with $\sum_i\lambda_i^2=1$,

$$\begin{aligned} Q_e(\lambda) &=-\frac\beta2+\beta\sum_{i<j}h_{ij}\lambda_i\lambda_j\\ &\le-\frac\beta2+\frac{\beta\mu_n(n-1)}2<0, \end{aligned}$$

where $\sum_{i<j}|\lambda_i\lambda_j|\le(n-1)/2$ follows from $(\sum_i|\lambda_i|)^2\le n$. Theorem 2.1 gives affine independence.

Every surviving cut separates at least one pair. Its squared edge length $\beta w_S$ is at most that pair's squared distance and hence at most $\beta(1+\mu_n)$. A box has squared circumradius one quarter of the sum of its squared edge lengths. There are at most $M$ of them, proving the radius estimate and the lemma.

Source precision.

The source refers to the 1990 near-regular embedding. Its original incidence proof and padding argument are preserved separately. The proof here supplies the all-$n$ array realization and stated dimension bound by a different elementary finite argument; it is not an author-issued correction. It avoids assuming a metric realization of a later residual array. Its explicit $\mu_n$ is sufficient, not claimed optimal.

Bears on. #174.