Open question: Separated sets in isotropic measures

[Note: This question is trivially impossible, though I leave the original form below. If \mu is the uniform measure on S^{k-1}, then any two sets of \Omega(1) measure are at distance O(1/\sqrt{k}) by concentration of measure. Thus it is impossible to find k sets satisfying the desired bound. I will try to think of the right question and repost. Unfortunately, I now recall having gone down this path a few times before, and I fear there may not be a relevant elementary open question after all.]

Here is an interesting and elementary (to state) open question whose resolution would give an optimal higher-order Cheeger esimate. The goal is to replace the factor {k^{O(1)}} in Theorem 1 of the previous post with a {\mathrm{polylog}(k)} factor, or even an optimal bound of {\sqrt{\log k}} .

Fix {k \in \mathbb N} and let {\mu} denote a probability measure on the {(k-1)} -dimensional sphere {S^{k-1}} . For the purpose of this question, one can assume that {\mu} is supported on a finite number of points. Suppose also that {\mu} is isotropic in the sense that, for every {\theta \in S^{k-1}} ,

\displaystyle  \int_{S^{k-1}} \langle x, \theta\rangle^2 \,d\mu(x)= \frac{1}{k}\,.

Now our goal is to find, for some {\varepsilon, \delta > 0} , a collection of (measurable) subsets {U_1, U_2, \ldots, U_k \subseteq S^{k-1}} such that {\mu(U_i) \geq \varepsilon/k} for each {i=1,2,\ldots,k} , and such that for {i \neq j} , we have

\displaystyle  \min_{x \in U_i, y \in U_j} \|x-y\|_2 \geq \delta\,.

In Lemma 5 of the previous post, we proved that this is possible for {\varepsilon \gtrsim 1} and {\delta \gtrsim k^{-5/2}} . In this paper, we improve the estimate on {\delta} somewhat, though the best-known is still {\delta \gtrsim k^{-c}} for some {c > 0} .

Is it possible to achieve {\varepsilon, \delta \gtrsim 1/\mathrm{poly}(\log k)} ? One could even hope for {\varepsilon \gtrsim 1} and {\delta \gtrsim 1/(\log k)} .

Two extreme cases are when (1) {\mu} is uniform on {S^{k-1}} , or (2) {\mu} is uniform on the standard basis {\{e_1,e_2,\ldots,e_k\}} . In the first case, one can easily achieve {\varepsilon,\delta \gtrsim 1} (in fact, one could even get {100k} sets satisfying these bounds). In the second case, taking each set to contain a single basis vector achieves {\varepsilon = 1} and {\delta = \sqrt{2}} .

If one only wants to find, say, {\lceil 3k/4\rceil} sets instead of {k} sets, then it is indeed possible to achieve {\varepsilon \gtrsim 1} and {\delta \gtrsim 1/O(\log k)} . [Update: This is actually impossible for the reasons stated above. To get the corresponding bounds, we do a random projection into O(\log k) dimensions. This only preserves the original Euclidean distances on average.] Furthermore, for {\varepsilon \gtrsim 1} , this bound on {\delta} is asymptotically the best possible.

3 thoughts on “Open question: Separated sets in isotropic measures

  1. How much slack is allowed in the isotropic condition? Suppose $\mu$ is supported on $x_1,\ldots,x_n$ and we don’t have $\sum_i x_ix_i^T = k^{-1}\cdot I$, but we do have, say, $\|\sum_i x_ix_i^T – k^{-1}\cdot I\| \le \gamma$. Is there some value of $\gamma>0$ that is still acceptable for your application?

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