Nonlinear $p$-stable projection conjecture for $1<p<2$

Let $1<p<2$, let $P\subset\mathbb{R}^D$ be a set of $N$ points, and let $F:\mathbb{R}^D\to\mathbb{R}^k$ have independent entries $F_{ij}$ distributed as standard $p$-stable random variables. Let $\rho$, $\epsilon$, and $k$ be as in the theorem above, and define

$$\mu(\lambda):=\mathbb{E}\,\xi(\lambda F_{11}).$$

Nonlinear $p$-stable projection conjecture. With probability at least $1-N^{-(c-2)}$, for all $x,y\in P$, if $\lVert x-y\rVert_p=O(\epsilon^2)$ then

$$(1-\epsilon)\mu(\lVert x-y\rVert_p)\leq \rho(F(x),F(y))\leq (1+\epsilon)\mu(\lVert x-y\rVert_p),$$

and if $\lVert x-y\rVert_p=\Omega(\epsilon^2)$ then

$$\mu\left(\frac{\lVert x-y\rVert_p}{1+\epsilon}\right)\leq \rho(F(x),F(y))\leq \mu\left((1+\epsilon)\lVert x-y\rVert_p\right).$$

This extends the theorem's nonlinear embedding statement from Cauchy, or $1$-stable, projections to $p$-stable projections for $1<p<2$, using the same metric and target-dimension regime.