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

Mirosław Baran, and Leokadia Bialas-Ciez. "Product property for capacities in $ℂ^{N}$." Annales Polonici Mathematici 106.1 (2012): 19-29. <http://eudml.org/doc/286221>.

@article{MirosławBaran2012,
abstract = {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.},
author = {Mirosław Baran, Leokadia Bialas-Ciez},
journal = {Annales Polonici Mathematici},
keywords = {logarithmic capacity; Siciak's extremal function; convex set; convex symmetric body},
language = {eng},
number = {1},
pages = {19-29},
title = {Product property for capacities in $ℂ^\{N\}$},
url = {http://eudml.org/doc/286221},
volume = {106},
year = {2012},
}

TY - JOUR
AU - Mirosław Baran
AU - Leokadia Bialas-Ciez
TI - Product property for capacities in $ℂ^{N}$
JO - Annales Polonici Mathematici
PY - 2012
VL - 106
IS - 1
SP - 19
EP - 29
AB - 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.
LA - eng
KW - logarithmic capacity; Siciak's extremal function; convex set; convex symmetric body
UR - http://eudml.org/doc/286221
ER -