Critical central sections of the cube

Ambrus, Gergely

doi:10.1090/proc/15955

Critical central sections of the cube
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by Gergely Ambrus PDF

Proc. Amer. Math. Soc. 150 (2022), 4463-4474 Request permission

Abstract:

We study the volume of central hyperplane sections of the cube. Using Fourier analytic and variational methods, we retrieve a geometric condition characterizing critical sections which, by entirely different methods, was recently proven by Ivanov and Tsiutsiurupa [Anal. Geom. Metr. Spaces 9 (2021), pp. 1-18]. Using this characterization result, we prove that critical central hyperplane sections in the 3-dimensional case are all diagonal to a (possibly lower dimensional) face of the cube, while in the 4-dimensional case, they are either diagonal to a face, or, up to permuting the coordinates and sign changes, perpendicular to the vector $(1,1,2,2)$. This shows the existence of non-diagonal critical central sections.

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