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Source. Published pp. 228–231, Lemma 3.9 and its proof. (canonical PDF).

As printed (p. 228): let $Z=\{z_1,\ldots,z_{d+1}\}$ be an arbitrary simplex with circumradius $\rho(Z)=\rho^Z$ and let $\vartheta>0$ be an arbitrary real. Then there is a simplex $V=\{v_1,\ldots,v_{d+1}\}$ with $\rho(V)\le\rho^Z\sqrt{1+\vartheta/8}$ which is $\alpha$-hyper Ramsey for $\alpha=(\rho^Z)^2(1+\vartheta/8)-\rho(V)^2$, and such that $|\|v_i-v_{i'}\|^2-\|z_i-z_{i'}\|^2|\le\vartheta$ for all $1\le i,i'\le d+1$ (inequality (17)). Since Definition 3.1 needs $\alpha>0$, the printed weak radius bound is completed below by a strict one.

Form proved here. Let $Z=\{z_1,\ldots,z_{d+1}\}$ be a simplex of positive dimension, with circumradius $\rho>0$, and let $\theta>0$. There is a simplex $V=\{v_1,\ldots,v_{d+1}\}$ such that

$$\bigl|\|v_i-v_j\|^2-\|z_i-z_j\|^2\bigr|\le\theta \quad(1\le i,j\le d+1),$$

and $V$ is $\alpha_V$-hyper-Ramsey for the strictly positive number

$$\alpha_V=\rho^2(1+\theta/8)-\rho(V)^2>0.$$

In particular $\rho(V)<\rho\sqrt{1+\theta/8}$. The proof is complete relative to the exact external inputs theorem_2_2 and lemma_2_3.

Proof.

Center $Z$ at its circumcenter and first divide its coordinates by $\rho$. Write $\Delta>0$ for the least distance between these normalized vertices. Choose a positive rational number $u=p/v$ so small that

$$u<\min\{\theta,\theta/\rho^2,1,4\Delta\}, \qquad \eta=u/16.$$

The strict rational choice both handles rescaling of squared distances and leaves room for a final radius enlargement. Apply Lemma 2.3 with $d$ and $\eta$, obtaining $s,k,a$ and a $d$-dimensional unit sphere in a subspace of $\mathbb R^s$. Map the normalized simplex isometrically to that subspace and choose spread vectors $y_1,\ldots,y_{d+1}$ within $\eta$ of its vertices. Both the original and spread vectors have norm one, so

$$\bigl|\|y_i-y_j\|^2-\|z_i/\rho-z_j/\rho\|^2\bigr| \le4(2\eta)=u/2.$$

Since $2\eta<\Delta$, these spread vectors are distinct. Hence their underlying $k$-sets are distinct members of the full family of $r=\binom sk\ge d+1$ sets. Fix their indices once and for all.

Put $q=k+1$ and

$$\omega=8v q^{r-1},\qquad l=t\omega,\qquad b=tp,\qquad n=ls+bq^r=tD,\qquad D=\omega s+pq^r,$$

for positive integers $t$. Then

$$\frac{b}{l}=\frac{u}{8q^{r-1}},\qquad \frac{n-ls}{lq}=\frac u8, \qquad \lambda=\frac bn=\frac pD>0.$$

All parameters of Lemma 3.5 are admissible. Apply its construction and select the $d+1$ rows associated with the chosen spread vectors. Their vectors are linearly independent, so form a simplex. Remark 3.8 shows that, as $t$ varies, this simplex has one fixed congruence class. Multiply it by $\rho$ and call the resulting fixed target $V$.

Lemma 3.5 gives squared-distance error at most $u/2$ from the spread vectors before scaling. Combining the two errors and restoring the factor $\rho$ gives error at most $\rho^2u<\theta$ from $Z$. Its selected vectors all have squared norm

$$R_0^2=\rho^2(1+u/8).$$

It remains to construct density witnesses for this fixed $V$. For each $n=tD$, let $\mathcal P_n$ be all ordered partitions with the part sizes in Lemma 3.5, and map a partition $A$ to the vector $w^A$ having value $\rho a_j/\sqrt l$ on its part $A_j$, where $a_0=0$. Let $H_n=\{w^A:A\in\mathcal P_n\}$. Every vector has norm $R_0$, and $0<|H_n|\le q^n<(q+1)^n$.

This map need not be injective. Group the labels $0,\ldots,k$ according to their common value $a_j$. For each distinct value $c$, put $L_c=\sum_{j:a_j=c}l_j$. Every vector in $H_n$ has exactly $L_c$ coordinates with value $\rho c/\sqrt l$. To recover its labeled partition, split those positions into the label parts of prescribed sizes. Thus every fiber has the same size

$$F_n=\frac{\prod_c L_c!}{\prod_{j=0}^k l_j!}.$$

It follows that every $K\subseteq H_n$ has an inverse image of exactly the same relative density in $\mathcal P_n$.

Let $M_n$ be the full joint intersection array of the selected $d+1$ constructed partitions. It is obtained by summing the other coordinates of the full $r$-row array. Every cell has size at least $b=\lambda n$; all one-row marginals are the common positive integers $l_j$. Theorem 2.2 therefore gives a fixed $0<\epsilon<1$, independent of $t$, such that every inverse image of density at least $(1-\epsilon)^n$ contains partitions with joint array $M_n$. Their vector images have exactly the same norms and pairwise squared distances as the selected vectors: each squared distance is the sum of $\rho^2(a_j-a_{j'})^2/l$ over the corresponding pair-label intersection. Hence their images contain a congruent copy of $V$. The constant-fiber calculation proves the required weak density threshold for $K$ itself, including when some $a_j$ vanish or coincide.

We have witnesses on $S(R_0,n)$ for every $n=tD$, with a fixed target, cardinality base and density base. Fact 3.10 extends them to every sufficiently large dimension, on the same sphere. Finally let

$$R_*^2=\rho^2(1+\theta/8)>R_0^2.$$

The radius enlargement in definitions places witnesses exactly on $S(R_*,m)$ in every sufficiently large $m$, adjusting the density exponent by a fixed factor. Since $V\subseteq S(R_0,n)$ in its original realization, $\rho(V)\le R_0<R_*$. Therefore $\alpha_V=R_*^2-\rho(V)^2>0$, as required.

Source precision.

The source normalizes the circumradius and assumes rational $\theta$ without spelling out the effect on absolute squared-distance error. The smaller rational $u$ and the final radius lift prove its exact stated conclusion for every real $\theta>0$ and every positive $\rho$. They also ensure distinct selected spread vectors and strictly positive slack. The source's “natural correspondence” (p. 230) between vectors and partitions is not necessarily one-to-one; the constant-fiber proof is necessary in that generality. Positive-dimensional simplices are the range here; singletons are handled directly by the main theorem.

Bears on. #174.