Polydisc slicing in $ℂ^n$

Krzysztof Oleszkiewicz; Aleksander Pełczyński

Polydisc slicing in $ℂ^{n}$

Krzysztof Oleszkiewicz; Aleksander Pełczyński

Studia Mathematica (2000)

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

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Abstract

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Let D be the unit disc in the complex plane ℂ. Then for every complex linear subspace H in

ℂ^{n}

of codimension 1,

v o l_{2 n - 2} (D^{n - 1}) \leq v o l_{2 n - 2} (H \cap D^{n}) \leq 2 v o l_{2 n - 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} + σ e_{k}

for some 1 ≤ j < k ≤ n and some σ ∈ ℂ with |σ| = 1. We identify

ℂ^{n}

with

ℝ^{2 n}

; by

v o l_{k} (\cdot)

we denote the usual k-dimensional volume in

ℝ^{2 n}

. The result is a complex counterpart of Ball’s [B1] result for cube slicing.

How to cite

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Oleszkiewicz, 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 -

References

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