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Inequalities on projected volumes

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  • UserZarko Randelovic (University of Cambridge)
  • ClockThursday 30 January 2020, 14:30-15:30
  • HouseMR12.

If you have a question about this talk, please contact Andrew Thomason.

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.

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