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[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 X \subset {\Bbb R}^d be a finite set of points, and let 0 < \epsilon < 1. We say that a finite set Y \subset {\Bbb R}^d is a weak \epsilon-net for X (with respect to convex bodies) if, whenever B is a convex body which is large in the sense that |B \cap X| > \epsilon |X|, then B contains at least one point of Y. (If Y is contained in X, we say that Y is a strong \epsilon-net for X with respect to convex bodies.)

For example, in one dimension, if X = \{1,\ldots,N\}, and Y = \{ \epsilon N, 2 \epsilon N, \ldots, k \epsilon N \} where k is the integer part of 1/\epsilon, then Y is a weak \epsilon-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 \epsilon-net of size as small as O(1/\epsilon). Strong \epsilon-nets are of importance in computational learning theory, and are fairly well understood via Vapnik-Chervonenkis (or VC) theory; however, the theory of weak \epsilon-nets is still not completely satisfactory.

One can ask what happens in higher dimensions, for instance when X is a discrete cube X = \{1,\ldots,N\}^d. It is not too hard to cook up \epsilon-nets of size O_d(1/\epsilon^d) (by using tools such as Minkowski’s theorem), but in fact one can create \epsilon-nets of size as small as O( \frac{1}{\epsilon} \log \frac{1}{\epsilon} ) simply by taking a random subset of X of this cardinality and observing that “up to errors of \epsilon“, the total number of essentially different ways a convex body can meet X grows at most polynomially in 1/\epsilon. (This is a very typical application of the probabilistic method.) On the other hand, since X can contain roughly 1/\epsilon disjoint convex bodies, each of which contains at least \epsilon of the points in X, we see that no \epsilon-net can have size much smaller than 1/\epsilon.

Now consider the situation in which X is now an arbitrary finite set, rather than a discrete cube. More precisely, let f(\epsilon,d) be the least number such that every finite set X possesses at least one weak \epsilon-net for X with respect to convex bodies of cardinality at most f(\epsilon,d). (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 \epsilon of any given territory.

  • Problem 1: For fixed d, what is the correct rate of growth of f as \epsilon \to 0?

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