Product property for capacities in $ℂ^{N}$

Mirosław Baran ; Leokadia Bialas-Ciez

Annales Polonici Mathematici, Tome 105 (2012), p. 19-29 / Harvested from The Polish Digital Mathematics Library

Résumé

The paper deals with logarithmic capacities, an important tool in pluripotential theory. We show that a class of capacities, which contains the L-capacity, has the following product property: $C_{ν}(E₁ × E₂) = min(C_{ν₁}(E₁),C_{ν₂}(E₂))$, where $E_{j}$ and $ν_{j}$ are respectively a compact set and a norm in $ℂ^{N_{j}}$ (j = 1,2), and ν is a norm in $ℂ^{N₁+N₂}$, ν = ν₁⊕ₚ ν₂ with some 1 ≤ p ≤ ∞. For a convex subset E of $ℂ^{N}$, denote by C(E) the standard L-capacity and by $ω_{E}$ the minimal width of E, that is, the minimal Euclidean distance between two supporting hyperplanes in $ℝ^{2N}$. We prove that $C(E) = ω_{E}/2$ for a ball E in $ℂ^{N}$, while $C(E) = ω_{E}/4$ if E is a convex symmetric body in $ℝ^{N}$. This gives a generalization of known formulas in ℂ. Moreover, we show by an example that the last equality is not true for an arbitrary convex body.

Détail

Publié le : 2012-01-01

Zbl 1254.32046

EUDML-ID : urn:eudml:doc:286221