Random points and lattice points in convex bodies

By Imre Bárány. 2008

84 citations recorded by the source corpus.

Assume K subset of R-d is a convex body and X is a (large) finite subset of K? How many convex polytopes are there whose vertices belong to X? Is there a typical shape of such polytopes? How well does the maximal such polytope (which is actually the convex hull of X) approximate K? We are interested in these questions mainly in two cases. The first is when X is a random sample of n uniform, independent points from K. In this case motivation comes from Sylvester's famous four-point problem and from the theory of random polytopes. The second case is when X = K boolean AND Z(d) where Z(d) is the lattice of integer points in R-d and the questions come from integer programming and geometry of numbers. Surprisingly (or not so surprisingly), the answers in the two cases are rather similar.

Subfields: Applied Mathematics, Computer Graphics and Computer-Aided Design, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/e673c6a0f8217244a30d95ce4496a9b3aeb9a410

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Zakhar Kabluchko
2. C. Thaele
3. Daniel Temesvari
4. Matthias Schulte
5. Günter Last
6. Giovanni Peccati

In Computer Graphics and Computer-Aided Design

1. C. Thaele
2. Zakhar Kabluchko
3. Daniel Temesvari

In Discrete Mathematics and Combinatorics

1. C. Thaele
2. Zakhar Kabluchko
3. Daniel Temesvari

Citations in the corpus

• Expected intrinsic volumes and facet numbers of random beta‐polytopes (2017, methodology): “[5] I. Bárány, Random points and lattice points in convex bodies, Bull.”
• Minimizing Human Intervention in Online Classification (2025, unknown): “This is the volume of a Macbeath region, see e.g. Bárány (2008), where they recall in their Lemma 4.3 that u ( x ) ≤ 2 v ( x ) .”