Sparsest-cut and negative-type metrics

Recall that the sparsest-cut value $\\beta$ is defined as:
\\[ \\beta = \\min_{\\varnothing\\neq S\\subseteq V}\\; \\frac{|\\delta(S,\\overline{S})|}{|S|\\cdot|\\overline{S}|} \\]

Arora-Rao-Vazirani: Key Theorem

Assume $x_1,\\dots,x_n\\in S^{d-1}= \\{x\\in\\mathbb{R}^d\\,|\\,\\|x\\|=1\\}$ are such that for all $i,j,k\\in[n]$ it holds that $\\angle x_ix_jx_k \\leq \\pi/2$, as well as:
\\[ \\frac{1}{\\binom{n}{2}}\\sum_{i<j}\\| x_i - x_j\\|^2 \\geq \\epsilon > 0 \\]
then there exist $S,T$ such that:
  1. $|S|=\\Omega(n)$,
  2. $|T|=\\Omega(n)$,
  3. $d(S,T)=\\Omega\\Big(\\frac{1}{\\sqrt{\\log n}}\\Big)$, and
  4. $d(x_i,x_j)=\\|x_i-x_j\\|^2$
and so forth until the end of proof. $\\square$