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Category: General Tobias Metzlaff - The Chromatic Number of R^n via optimization of multivariate Chebyshev polynomials

21 July 2021

The chromatic number of R^n for a polytope norm is the minimal number of colors required for the corresponding infinite distance graph and is known to be at most 2^n. A lower spectral bound can be obtained by minimizing the Fourier transformation of a discrete measure, which is supported on the polytopes boundary. Under symmetry assumptions, the support of the measure consists of the orbits of a root system's weights by the action of the associated Weyl group and the polytope can be related to the corresponding fundamental domain. From the theory of root systems, a definition of multivariate Chebyshev polynomials of the first kind arises and we rewrite the spectral bound in terms of polynomial optimization. The domain of optimization is the image of a function, which is invariant under the Weyl group. We give a polynomial expression for this function depending on the type of the root system and describe the image as a basic semi algebraic set. Based on joint work with Evelyne Hubert, Philippe Moustrou and Cordian Riener.

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