# Inequalities on projected volumes

• Zarko Randelovic (University of Cambridge)
• Thursday 30 January 2020, 14:30-15:30
• MR12.

Given $2n – 1$ real numbers $x_A$ indexed by the non-empty subsets $A \subset \{ 1,\ldots,n \}$, is it possible to construct a body $T$ in $Rn$ such that $x_A = \| T_A \|$, where $\| T_A \|$ is the $\| A \|$-dimensional volume of the projection of $T$ onto the subspace spanned by the axes of $A$? As it is more convenient to take logarithms, we denote by $\psi_n$ the set of all vectors $x$ for which there is a body $T$ such that $x_A = \log \| T_A \|$ for all $A$. Bollob\’as and Thomason showed that $\psi_n$ is containd in the polyhedral cone defined by the class of uniform cover inequalities’. Tao and Zeng conjectured that the convex hull of $\psi_n$ is equal to the cone given by the uniform cover inequalities.

We show that this conjecture is not right, but is nearly’ right.

Joint work with Imre Leader and Eero Raty.

This talk is part of the Combinatorics Seminar series.