You are currently browsing the tag archive for the ‘convex geometry’ tag.
[This post is authored by Gil Kalai, who has kindly “guest blogged” this week’s “open problem of the week”. – T.]
This is a problem in discrete and convex geometry. It seeks to quantify the intuitively obvious fact that large convex bodies are so “fat” that they cannot avoid “detection” by a small number of observation points. More precisely, we fix a dimension d and make the following definition (introduced by Haussler and Welzl):
- Definition: Let
be a finite set of points, and let
. We say that a finite set
is a weak
-net for X (with respect to convex bodies) if, whenever B is a convex body which is large in the sense that
, then B contains at least one point of Y. (If Y is contained in X, we say that Y is a strong
-net for X with respect to convex bodies.)
For example, in one dimension, if , and
where k is the integer part of
, then Y is a weak
-net for X with respect to convex bodies. Thus we see that even when the original set X is very large, one can create a
-net of size as small as
. Strong
-nets are of importance in computational learning theory, and are fairly well understood via Vapnik-Chervonenkis (or VC) theory; however, the theory of weak
-nets is still not completely satisfactory.
One can ask what happens in higher dimensions, for instance when X is a discrete cube . It is not too hard to cook up
-nets of size
(by using tools such as Minkowski’s theorem), but in fact one can create
-nets of size as small as
simply by taking a random subset of X of this cardinality and observing that “up to errors of
“, the total number of essentially different ways a convex body can meet X grows at most polynomially in
. (This is a very typical application of the probabilistic method.) On the other hand, since X can contain roughly
disjoint convex bodies, each of which contains at least
of the points in X, we see that no
-net can have size much smaller than
.
Now consider the situation in which X is now an arbitrary finite set, rather than a discrete cube. More precisely, let be the least number such that every finite set X possesses at least one weak
-net for X with respect to convex bodies of cardinality at most
. (One can also replace the finite set X with an arbitrary probability measure; the two formulations are equivalent.) Informally, f is the least number of “guards” one needs to place to prevent a convex body from covering more than
of any given territory.
- Problem 1: For fixed d, what is the correct rate of growth of f as
?
Recent Comments