My question is very simple:
Given $d\ge 3$, does there exist $s\in (0,1)$ and an embedding $f:S^{d-1}\to \mathbb{R}^d$ such that $$ |f(x)-f(y)| \ge |x-y|^s \quad\textrm{if } |x-y|<r, $$ for some $r>0$?
In other words, I'm looking for some kind of "snowflaking" (or "snowballing") of the Euclidean sphere. Some comments:
It is very important for me that the embedding is into $\mathbb{R}^d$; Assouad's embedding Theorem gives such an embedding into some $\mathbb{R}^D$ for some large $D=D(d)$.
For $d=2$, it is easy to see that the answer is affirmative: for example, there exists such an $f$ mapping the circle onto the Von Koch snowflake.
The closest I've found to an answer is the impressive paper "Reifenberg flat metric spaces, snowballs, and embeddings" by G. David and T. Toro (available here behind a paywall). There they prove that for a class of $d-1$-dimensional (in some sense) metric spaces $X$, including hyperplanes in $\mathbb{R}^d$ and small perturbations of them, there exist some $s<1, C>1, r>0$ and an embedding $f:X\to\mathbb{R}^d$ such that $$ C^{-1} d(x,y)^s \le |f(x)-f(y)| \le C d(x,y)^s\quad\textrm{if } d(x,y)<r. $$ The sphere $S^{d-1}$ does not meet their conditions, and their methods appear to be local in some sense. On the other hand, I only need the lower bound, so it is possible that there is a much simpler construction.
Motivation: we are studying random fractal sets of the form $\mathbb{R}^d\setminus\bigcup_i\Lambda_i$, where the $\Lambda_i$ are chosen according to a Poisson point process taking values in compact subsets of $\mathbb{R}^d$. If the (boundaries of) $\Lambda_i$ are embeddings of the sphere satisfying the condition in the question, then the random fractal will have many desirable geometric properties, which we don't know how to achieve in any other way.
