# Polydisc slicing in ${\u2102}^{n}$

Krzysztof Oleszkiewicz; Aleksander Pełczyński

Studia Mathematica (2000)

- Volume: 142, Issue: 3, page 281-294
- ISSN: 0039-3223

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topOleszkiewicz, Krzysztof, and Pełczyński, Aleksander. "Polydisc slicing in $ℂ^n$." Studia Mathematica 142.3 (2000): 281-294. <http://eudml.org/doc/216804>.

@article{Oleszkiewicz2000,

abstract = {Let D be the unit disc in the complex plane ℂ. Then for every complex linear subspace H in $ℂ^n$ of codimension 1, $vol_\{2n-2\}(D^\{n-1\}) ≤ vol_\{2n-2\}(H ∩ D^\{n\}) ≤ 2vol_\{2n-2\}(D^\{n-1\})$. The lower bound is attained if and only if H is orthogonal to the versor $e_\{j\}$ of the jth coordinate axis for some j = 1,...,n; the upper bound is attained if and only if H is orthogonal to a vector $e_\{j\} + σe_\{k\}$ for some 1 ≤ j < k ≤ n and some σ ∈ ℂ with |σ| = 1. We identify $ℂ^n$ with $ℝ^\{2n\}$; by $vol_\{k\}(·)$ we denote the usual k-dimensional volume in $ℝ^\{2n\}$. The result is a complex counterpart of Ball’s [B1] result for cube slicing.},

author = {Oleszkiewicz, Krzysztof, Pełczyński, Aleksander},

journal = {Studia Mathematica},

keywords = {volume of section; Bessel functions; polydisc},

language = {eng},

number = {3},

pages = {281-294},

title = {Polydisc slicing in $ℂ^n$},

url = {http://eudml.org/doc/216804},

volume = {142},

year = {2000},

}

TY - JOUR

AU - Oleszkiewicz, Krzysztof

AU - Pełczyński, Aleksander

TI - Polydisc slicing in $ℂ^n$

JO - Studia Mathematica

PY - 2000

VL - 142

IS - 3

SP - 281

EP - 294

AB - Let D be the unit disc in the complex plane ℂ. Then for every complex linear subspace H in $ℂ^n$ of codimension 1, $vol_{2n-2}(D^{n-1}) ≤ vol_{2n-2}(H ∩ D^{n}) ≤ 2vol_{2n-2}(D^{n-1})$. The lower bound is attained if and only if H is orthogonal to the versor $e_{j}$ of the jth coordinate axis for some j = 1,...,n; the upper bound is attained if and only if H is orthogonal to a vector $e_{j} + σe_{k}$ for some 1 ≤ j < k ≤ n and some σ ∈ ℂ with |σ| = 1. We identify $ℂ^n$ with $ℝ^{2n}$; by $vol_{k}(·)$ we denote the usual k-dimensional volume in $ℝ^{2n}$. The result is a complex counterpart of Ball’s [B1] result for cube slicing.

LA - eng

KW - volume of section; Bessel functions; polydisc

UR - http://eudml.org/doc/216804

ER -

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