Skip to main content
SpringerLink
Log in

Volume Formulas in Lp-Spaces

  • Published:
Positivity Aims and scope Submit manuscript

Abstract

We extend classical volume formulas for ellipsoids and zonoids to p-sums of segments

$${vol}\left( {\sum\limits_{i=1}^m { \oplus_p } [ -x_i ,x_i ]} \right)^{1/n} \sim_{c_p} n^{ - \frac{1}{{p'}}} \left( {\sum\limits_{card(I) = n} {|\det (x_i)_i |^p}} \right)^{\frac{1}{{pn}}}$$

where x1,...,xm are m vectors in

\(\mathbb{R}^n ,\frac{1}{p} + \frac{1}{{p\prime }} = 1\).

According to the definition of Firey, the Minkowski p-sum of segments is given by

$$\sum\limits_{i = 1}^m { \oplus _p [ - x_{i,} x_i ]} = \left\{ {\sum\limits_{i = 1}^m {\alpha _i } x_i \left| {\left( {\sum\limits_{i = 1}^m {|\alpha _i |^{p^\prime } } } \right)} \right.^{\frac{1}{{p^\prime }}} \leqslant 1} \right\}.$$

We describe related geometric properties of the Lewis maps associated to classical operator norms.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Ball, K.: 1991, ‘Normed spaces with a weak Gordon-Lewis property’, In Functional Analysis. Springer Lectures Notes 1470.

  • Ball, K. and A. Pajor: 1990, Convex bodies with few faces. Proc. of the AMS 110, 225–231.

    Google Scholar 

  • Barany, I. and Z. Furedi: 1987, ‘Computing volume is difficult’. Discrete Comput. Geom. 2, 319–326.

    Google Scholar 

  • Barthe, F.: 1995, ‘Mesure unimodales et sections des Boules B np ’. C.R. Acad. Sci. Paris t 321 Serie I, 865–868.

  • Bourgain, J. and V.D. Milman: 1987, ‘New volume ratio properties for convex symmetric bodies in ℝn’. Invent. Math. 88, 319–340.

    Google Scholar 

  • Burago, Y. and V. Zalgaller: 1980, ‘Geometric Inequalities’, “Nauka”, Leningrad, 1980, 288 pp. (Russian) English translation: 1988, Springer Verlag.

  • Caetano, A.: 1992, ‘Weyl numbers in sequence spaces and sections of unit balls’. J. of Funct. Analysis 106, 1–17.

    Google Scholar 

  • Carl, B. and A. Pajor: 1988, ‘Gelfand numbers of operators with values in a Hilbert space’. Invent. Math. 94, 479–504.

    Google Scholar 

  • Defant, A. and M. Junge: ‘On some matrix inequalities in Banach spaces’. preprint.

  • Dudley, R.M.: 1967, ‘The sizes of compact subsets of Hilbert space and continuity of Gaussian processes’. J. Funct. Anal. 1, 290–330.

    Google Scholar 

  • Firey, W.J.: 1962, ‘The mixed area of convex bodies’. Math. Scand 10, 17–24.

    Google Scholar 

  • Geiss, S.: 1990, ‘Grothendieck numbers of Linear and continuous operators on Banach spaces’. Math. Nachr. 148, 65–79.

    Google Scholar 

  • Geiss, S.: 1992, ‘Antisymmetric Tensor Products of Absolutely p-summing Operators’, J. Approximation Theory 68, 223–246.

    Google Scholar 

  • Gluskin, E.D.: 1989, ‘Extremal properties and orthogonal parallelepipeds and their applications to the geometry of Banach spaces’. Math. USSR Sbornik 64(1), 85–96.

    Google Scholar 

  • Garling, D.J.H. and Y. Gordon: 1971, Relations between some constants associated with finitedimensional Banach spaces. Israel J. Math. 9, 346–361.

    Google Scholar 

  • Gordon, Y. and M. Junge: Volume ratios in Lp-spaces. preprint.

  • Gordon, Y. and N.J. Kalton: 1994, ‘Local structure for quasi-normed spaces’. Bull. Sci. Math. 118, 441–453.

    Google Scholar 

  • Gordon, Y. and D.R. Lewis: 1974, ‘Absolutely summing operators and local unconditional structures’. Acta Math. 133, 27–48.

    Google Scholar 

  • Gordon, Y. and D.R. Lewis: 1975, ‘Banach ideals on Hilbert spaces’. Studia Math. 54, 161–172.

    Google Scholar 

  • Gordon, Y.: 1966, ‘On the projection and MacPhail constants in ℓ np ’. Israel J. Math. 4, 177–188.

    Google Scholar 

  • Gordon, Y., M. Meyer and A. Pajor: 1996, ‘Ratios of volumes and factorization through ∞’. Ill. Journal of Math. 40(1), 91–107.

    Google Scholar 

  • Gruber P.M. and C.G. Lekkerkerker: 1987, Geometry of numbers, vol. 37 North Holland Publ. Co.

  • John, F.: 1948, ‘Extremum problems with inequalities as subsidiary conditions’. Courant Anniversary volume, Interscience, New York, 187–204.

    Google Scholar 

  • Junge, J.: 1989, ‘Charakterisierung der K-Konvexität durch Volumenquotienten’. Master thesis, Kiel.

  • Junge, J.: 1994, ‘Hyperplane conjecture for quotient spaces of Lp’. Forum Math. 6, 617–635.

    Google Scholar 

  • Kwapieή S. and W.A. Woyczyήski: 1992, Random Series and Stochastic Integrals. Birkhäuser.

  • Lewis, D.: 1976, ‘Ellipsoids defined by Banach ideal norms’. Mathematika 26, 18–29.

    Google Scholar 

  • Mascioni, V.: 1991, ‘On generalized volume ratio numbers’. Bull. Sc.Math., 2nd serie, 115, 453–510.

    Google Scholar 

  • Meyer, M. and A. Pajor: 1988, ‘Sections of the unit ball of ℓ np ’. J. of Funct. Anal. 80, 109–123.

    Google Scholar 

  • Pajor, A. and N. Tomczak-Jaegermann: 1989, ‘Volume ratio and other s-numbers related to local properties of Banach spaces’. J. of Funct Anal. 87, 273–293.

    Google Scholar 

  • Pelczynski, A. and S.J. Szarek: 1991, ‘On parellelepipieds of minimal volume containing a convex symmetric body in ℝn’. Math. Proc. Camb. Phil. Soc., 109–125.

  • Pietsch, A.: 1980, Operators Ideals. Amsterdam, North Holland Pub. Co.

    Google Scholar 

  • Pisier, G.: 1982, ‘Holomorphic semi groups and the geometry of Banach spaces’. Anals of Math. 115, 326–392.

    Google Scholar 

  • Pisier, G.: 1986, ‘Factorization of linear operators and the Geometry of Banach spaces’. CBMS (Regional conferences of the A.M.S.) 60. Reprinted with correction 1987.

  • Pisier, G.: 1989, ‘The Volume of Convex Bodies and Banach Space Geometry’. Cambridge Tracts in Mathematics 94, Cambridge Univ. Press.

  • Santalò, L.: 1949, ‘Un invarianta afin parap los cuerpos convexos del espacio de n dimensiones’. Portugal. Math. 8, 155–161.

    Google Scholar 

  • Schütt, C.: 1982, ‘On the volume of unit balls in Banach spaces’. Compositio Math. 47, 393–407.

    Google Scholar 

  • Tomczak-Jaegermann, N.: 1989, ‘Banach-Mazur distances and finite-dimensional operator ideals’. Pitman Monographs and surveys in pure and applied Mathematics. Longman Scientific & Technical, Wiley, New York.

    Google Scholar 

  • Vaaler, G.: 1979, ‘A geometric inequality with applications to linear forms’. Pacific J. of Math. 83, 543–553.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dept. of Mathematics, Technion, Haifa, 32000, Israel

    Yehoram Gordon

  2. Math. Seminar, Christian-Albrechts Universität, Ludewig-Meyn-Str. 4, 24098, Kiel, Germany

    Marius Junge

Authors
  1. Yehoram Gordon
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marius Junge
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gordon, Y., Junge, M. Volume Formulas in Lp-Spaces. Positivity 1, 7–43 (1997). https://doi.org/10.1023/A:1009731300757

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1009731300757

Search

Navigation