Résumé : It is known that the largest possible volume for the intersection of a d-dimensional unit hypercube with a hyperplane H through its center is the square root of 2. This happens when H is orthogonal to a diagonal of a square face of the hypercube. This question can be generalized by considering the hyperplanes H at a fixed distance t to the center of the hypercube. Vitali Milman asked which among these hyperplanes have an intersection of maximal volume with the hypercube and conjectured that this happens when H is orthogonal to a diagonal or a sub-diagonal of the hypercube, depending on the value of t. Several recent results on this question are presented in this talk. In particular, when t is large enough (and smaller than the circumradius of the hypercube), the maximal volume is achieved exactly when the hyperplane is orthogonal to a diagonal of the hypercube. For smaller values of t, local extremality results will be presented at the diagonals and sub-diagonals of the hypercube.

Dernière modification : Thursday 21 November 2024

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