Volume of sections of high dimensional convex bodies

• Applicant: Professor Dr. Hermann König
• Subject Area: Mathematics
• Term: from 2012 to 2016
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 227612601

The volumes of hyperplane sections and slabs of high dimensional convex bodies and the dual question about volumes of projections have been studied intensively in the last 30 years. In the deterministic case, the central object of research were convex bodies, which are unit balls of classical norms, like the lp-unit ball, or standard bodies like the simplex in Rn. The determination of hyperplanes which give minimal and maximal volumes of sections and slabs has only been successful in special cases even in this standard situation. The reason to study these volumes were amongst other things the (by now solved) Busemann- Petty problem and the (in general still) unsolved hyperplane conjecture. In this project we will study minimal and maximal volumes of sections and slabs in lp-unit balls, in particular for p = ∞, large values of p and for p = 1 and in the simplex for open cases. Results in the two- or three-dimensional situation cannot be carried over, in general, to the n-dimensional case (for large dimensions n). The methods used so far come from Functional Analysis, Fourier Analysis and Convex Geometry. There are intrinsic relations with classical inequalities, for example the Khintchine inequality.

• DFG Programme: Research Grants