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## Studia Mathematica

2000 | 142 | 3 | 281-294
Tytuł artykułu

### Polydisc slicing in $ℂ^n$

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
EN
Czasopismo
Rocznik
Tom
Numer
Strony
281-294
Opis fizyczny
Daty
wydano
2000
otrzymano
2000-01-14
poprawiono
2000-05-17
poprawiono
2000-08-16
Twórcy
• Institute of Mathematics, Warsaw University, Banacha 2, 02-097 Warszawa, Poland
• Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-950 Warszawa, Poland
Bibliografia
• [B1] K. Ball, Cube slicing in $ℝ^n$, Proc. Amer. Math. Soc. 97 (1986), 465-473.
• [B2] K. Ball, Volumes of sections of cubes and related problems, in: Geometric Aspects of Functional Analysis (1987-88), Lecture Notes in Math. 1376, Springer, Berlin, 1989, 251-260.
• [Bar] F. Barthe, On a reverse form of the Brascamp-Lieb inequality, Invent. Math. 134 (1998), 335-361.
• [F] W. Feller, Introduction to Probability Theory and its Applications, Vol. II, Wiley, New York, 1996.
• [H] D. Hensley, Slicing the cube in $R^n$ and probability (bounds for the measure of a central cube slice in $R^n$ by probability methods), Proc. Amer. Math. Soc. 73 (1979), 95-100.
• [KL] A. Koldobsky and M. Lifshits, Average volume of sections of star bodies, in: Geometric Aspects of Functional Analysis, Israel Seminar (GAFA), 1996-2000, V. Milman and G. Schechtman (eds.), Lecture Notes in Math. 1745, Springer, 2000, 119-146.
• [K] H. König, On the best constants in the Khintchine inequality for variables on the spheres, preprint, Kiel Universität, 1998.
• [KK] H. König and S. Kwapień, Best Khintchine type inequalities for sums of independent, rotationally invariant random vectors, Positivity, to appear.
• [MP] M. Meyer and A. Pajor, Sections of the unit ball of $l_p^n$, J. Funct. Anal. 80 (1988), 109-123.
• [NP] F. L. Nazarov and A. N. Podkorytov, Ball, Haagerup, and distribution functions, in: Complex Analysis, Operator Theory, and Related Topics: S. A. Vinogradov - In Memoriam, V. Havin and N. Nikolski (eds.), Oper. Theory Adv. Appl. 113, Birkhäuser, Basel, 2000, 247-268.
• [P] G. Pólya, Berechnung eines bestimmten Integrals, Math. Ann. 74 (1913), 204-212.
• [RG] I. M. Ryshik and I. S. Gradstein, Tables of Series, Products and Integrals, Deutscher Verlag Wiss., Berlin, 1957.
• [S] M. Schmuckenschläger, Volume of intersections and sections of the unit ball of $l_p^n$, Proc. Amer. Math. Soc. 126 (1998), 1527-1530.
• [V] J. D. Vaaler, A geometric inequality with applications to linear forms, Pacific J. Math. 83 (1979), 543-553.
• [W] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Univ. Press, Cambridge, 1995.
Typ dokumentu
Bibliografia
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