Polydisc slicing in $ℂ^n$

Oleszkiewicz, Krzysztof; Pełczyński, Aleksander

Polydisc slicing in

ℂ^{n}

Oleszkiewicz, Krzysztof ; Pełczyński, Aleksander

Studia Mathematica, Tome 141 (2000), p. 281-294 / Harvested from The Polish Digital Mathematics Library

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Résumé

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.

Publié le : 2000-01-01

Zbl 0971.32018

EUDML-ID : urn:eudml:doc:216804

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     title = {Polydisc slicing in $$\mathbb{C}$^n$
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     journal = {Studia Mathematica},
     volume = {141},
     year = {2000},
     pages = {281-294},
     zbl = {0971.32018},
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Oleszkiewicz, Krzysztof; Pełczyński, Aleksander. Polydisc slicing in $ℂ^n$
            . Studia Mathematica, Tome 141 (2000) pp. 281-294. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv142i3p281bwm/

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