imrebarany

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  • (KMRS Seminar) Imre Barany, Random points and lattice points in convex bodies

    FYI (KMRS Seminar)

    Random points and lattice points in convex bodies
    Imre Barany
    Hungarian Academy of Sciences & University College London
    2014/08/25-08/26 Monday & Tuesday
    4:00PM – 5:00PM Room 1409
    Assume K is a convex body in R^d and X is a (large) finite subset of K. How many convex polytopes are there whose vertices belong toX? 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? In this lecture I will talk about 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 is the set of lattice points contained in K 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. The methods are, however, very different.
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  • Imre Bárány; Tensors, colours, octahedra
    Tensors, colours, octahedra
    Imre Bárány
    Alfréd Rényi Mathematical Institute
    Hungarian Academy of Sciences
    and
    University College London
    2013/04/26 Fri 4PM-5PM – ROOM 3433
    Several classical results in convexity, like the theorems of Caratheodory, Helly, and Tverberg, have colourful versions. In this talk I plan to explain how two methods, the octahedral construction and Sarkaria’s tensor trick, can be used to prove further extensions and generalizations of such colourful theorems.
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