Colloquium by Ansgar Freyer (Technische Universität Berlin): Shaking a convex body in order to count its lattice points
We prove inequalities on the number of lattice points inside a convex body K in terms of its volume and its successive minima. The successive minima of a convex body have been introduced by Minkowski and since then, they play a major role in the geometry of numbers.
A key step in the proof is a technique from convex geometry known as Blascke's shaking procedure by which the problem can be reduced to anti-blocking bodies, i.e., convex bodies that are "located in the corner of the positive orthant".
As a corollary of our result, we will obtain an upper bound on the number of lattice points in K in terms of the successive minima, which is equivalent to Minkowski's Second Theorem, giving a partial answer to a conjecture by Betke et al. from 1993.
This is a joint work with Eduardo Lucas Marín.
Time & Location
Jul 05, 2021 | 04:00 PM s.t.