A partition P of Rd is called a
(k,ε)-secluded partition if, for every p∈Rd,
the ball B∞(ε,p) intersects at most k
members of P. A goal in designing such secluded partitions is to
minimize k while making ε as large as possible. This partition
problem has connections to a diverse range of topics, including deterministic
rounding schemes, pseudodeterminism, replicability, as well as Sperner/KKM-type
results.
In this work, we establish near-optimal relationships between k and
ε. We show that, for any bounded measure partitions and for any
d≥1, it must be that k≥(1+2ε)d. Thus, when k=k(d) is
restricted to poly(d), it follows that ε=ε(d)∈O(dlnd). This bound is tight up to log factors, as it is
known that there exist secluded partitions with k(d)=d+1 and
ε(d)=2d1. We also provide new constructions of secluded
partitions that work for a broad spectrum of k(d) and ε(d)
parameters. Specifically, we prove that, for any
f:N→N, there is a secluded partition with
k(d)=(f(d)+1)⌈f(d)d⌉ and
ε(d)=2f(d)1. These new partitions are optimal up to
O(logd) factors for various choices of k(d) and ε(d). Based
on the lower bound result, we establish a new neighborhood version of Sperner's
lemma over hypercubes, which is of independent interest. In addition, we prove
a no-free-lunch theorem about the limitations of rounding schemes in the
context of pseudodeterministic/replicable algorithms