Let S be a subset of Rd with finite positive Lebesgue measure.
The Beer index of convexity b(S) of S is the probability
that two points of S chosen uniformly independently at random see each other
in S. The convexity ratio c(S) of S is the Lebesgue
measure of the largest convex subset of S divided by the Lebesgue measure of
S. We investigate the relationship between these two natural measures of
convexity.
We show that every set S⊆R2 with simply connected
components satisfies b(S)≤αc(S) for an
absolute constant α, provided b(S) is defined. This
implies an affirmative answer to the conjecture of Cabello et al. that this
estimate holds for simple polygons.
We also consider higher-order generalizations of b(S). For
1≤k≤d, the k-index of convexity bk(S) of a set
S⊆Rd is the probability that the convex hull of a
(k+1)-tuple of points chosen uniformly independently at random from S is
contained in S. We show that for every d≥2 there is a constant
β(d)>0 such that every set S⊆Rd satisfies
bd(S)≤βc(S), provided
bd(S) exists. We provide an almost matching lower bound by
showing that there is a constant γ(d)>0 such that for every
ε∈(0,1) there is a set S⊆Rd of Lebesgue
measure 1 satisfying c(S)≤ε and
bd(S)≥γlog21/εε≥γlog21/c(S)c(S).Comment: Final version, minor revisio