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## Annales Polonici Mathematici

2012 | 106 | 1 | 19-29
Tytuł artykułu

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

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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.
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19-29
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2012
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• Institute of Mathematics, Faculty of Mathematics and Computer Science, Jagiellonian University, 30-348 Kraków, Poland
autor
• Institute of Mathematics, Faculty of Mathematics and Computer Science, Jagiellonian University, 30-348 Kraków, Poland
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