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Дни комбинаторики и геометрии II
15 апреля 2020 г. 15:00, Онлайн-конференция


On the volume of sections of the cube

Г. М. Иванов


https://youtu.be/NWp3lFBS4TE

Аннотация: The problem of volume extrema of the intersection of the standard $n$-dimensional cube $\square^n=[-1,1]^n$ with a $k$-dimensional linear subspace $H$ has been studied intensively. The celebrated Vaaler theorem says that only the coordinate subspaces are the volume minimizers. Using the Brascamb-Lieb inequality, K. Ball proved two upper bounds which are tight for some $k$ and $n$. Typically, methods of functional analysis or some tricky inequalities for measures are used in such problems. In this talk, we will discuss a 'naive' variational principle for the problem of volume extrema of $\square^n \cap H$ and some geometrical consequences of this principle. Particularly, we will sketch how to find all planar maximizers ($k=2$). Planar maximizers were unknown for all odd $k$ starting with $5$.

Joint work with Igor Tsiutsiurupa.


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