# Lecture 2, by Imre Bárány (Alfréd Rényi Mathematical Institute of the Hungarian Academy of Science, Budapest): Cells in the box and a hyperplane

It is well known that a line can intersect at most 2*n*−1 cells of the *n*×*n* chessboard. What happens in higher dimensions: how many cells of the *d*-dimensional [0,*n*]^^{d} box can a hyperplane intersect? We answer this question asymptotically. We also prove the integer analogue of the following fact. If *K,L* are convex bodies in * R^d* and

*K*⊂

*L*, then the surface area

*K*is smaller than that of

*L*. This is joint work with Péter Frankl.

### Time & Location

May 30, 2022 | 04:00 PM s.t.

Chemistry building

Arnimallee 22

14195 Berlin

Hörsaal A

Campus map