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

• Autorzy: Mirosław Baran , Leokadia Bialas-Ciez
• Języki publikacji: EN

Abstrakty

EN

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.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 2012
• Tom: 106
• Numer: 1
• Strony: 19-29

Daty

wydano

2012

Twórcy

autor

Mirosław Baran

• Institute of Mathematics, Faculty of Mathematics and Computer Science, Jagiellonian University, 30-348 Kraków, Poland

autor

Leokadia Bialas-Ciez

• Institute of Mathematics, Faculty of Mathematics and Computer Science, Jagiellonian University, 30-348 Kraków, Poland

Identyfikatory

DOI

10.4064/ap106-0-2