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Given a convex compact set K C [~2 what is the largest n such that K contains a convex lattice n-gon? We answer this question asymptotically. It turns out that the maxima[ n is related to the largest affine perimeter that a convex set contained in K can have. This, in turn, gives a new characterization of /4o, the convex set in K having maximal affine… (More)
What is the minimum perimeter of a convex lattice n-gon? This question was answered by Jarník in 1926. We solve the same question, and prove a limit shape result, in the case when perimeter is measured by a (not necessarily symmetric) norm.
In 1979 Vaaler proved that every d-dimensional central section of the cube [−1, 1] n has volume at least 2 d. We prove the following sharp combinatorial analogue. Let K be a d-dimensional subspace of R n. Then, there is a probability measure P on the section [−1, 1] n ∩ K, so that the quadratic form [−1,1] n ∩K v ⊗ v dP (v) dominates the identity on K (in… (More)
What is the minimum perimeter of a convex lattice n-gon? This question was answered by Jarník in 1926. We solve the same question in the case when perimeter is measured by a (not necessarily symmetric) norm.
The workshop on Discrete Geometry was attended by 53 participants , many of them young researchers. In 13 survey talks an overview of recent developments in Discrete Geometry was given. These talks were supplemented by 16 shorter talks in the afternoon, an open problem session and two special sessions.