Relating transfinite diameters using an Okounkov body. (English) Zbl 1435.32045

This paper studies the notion of transfinite diameter using an Okounkov body. For a given collection of \(m+1\) points \(F:=\{\zeta_0,\dots,\zeta_m\}\subset K\subset\mathbb{C}\), let \(\mathrm{VDM}(F):=\prod_{j<k}(\zeta_j-\zeta_k)\). Maximizing \(|\mathrm{VDM}(F)|^{1/m}\) over all subsets \(F\) of size \(m+1\) gives \(d_m(K)\), the \(m\)th-order diameter of \(K\). A subset for which the maximum is attained is called a set of Fekete points of order \(m\), and the transfinite diameter is \(d(K):=\lim_{m\to\infty}d_m(K)\).
An Okounkov body is a well-studied object in complex geometry that gives a simplified model of the local behaviour of a holomorphic line bundle over a complex manifold. The main result of this paper is given in Theorem 5.8, in which relations between the transfinite diameter of a locally circled subset \(K\) of the complexified sphere in \(\mathbb{C}^3\) and notions of weighted transfinite diameter of the projection of \(K\) to \(\mathbb{C}^2\) are derived. The method is based on connecting a Chebyshev transform of \(K\) on an Okounkov body of \(V\) to the classical Chebyshev transform of the projection of \(K\) to \(\mathbb{C}^2\).


32U20 Capacity theory and generalizations
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
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