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Non-zero boundaries of Leibniz half-spaces

Author: Fuchang Gao
Journal: Proc. Amer. Math. Soc. 133 (2005), 1757-1762
MSC (2000): Primary 46B20
Published electronically: November 19, 2004
MathSciNet review: 2120275
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Abstract: It is proved that for any $d\geq 3$, there exists a norm $\Vert\cdot\Vert$ and two points $a$, $b$ in $\mathbb{R}^d$ such that the boundary of the Leibniz half-space $H(a,b)=\{x\in \mathbb{R}^d: \Vert x-a\Vert\leq\Vert x-b\Vert\}$ has non-zero Lebesgue measure. When $d=2$, it is known that the boundary must have zero Lebesgue measure.

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  • 1. Keith Ball and Alain Pajor, The entropy of convex bodies with ``few'' extreme points, Geometry of Banach spaces (Strobl, 1989), London Math. Soc. Lecture Note Ser., vol. 158, Cambridge Univ. Press, Cambridge, 1990, pp. 25-32. MR 93b:46024
  • 2. Bernd Carl, Entropy numbers of diagonal operators with an application to eigenvalue problems, J. Approx. Theory 32 (1981), no. 2, 135-150. MR 83a:47024
  • 3. Bernd Carl, Ioanna Kyrezi, and Alain Pajor, Metric entropy of convex hulls in Banach spaces, J. London Math. Soc. (2) 60 (1999), no. 3, 871-896. MR 2001c:46019
  • 4. Bernd Carl and Irmtraud Stephani, Entropy, compactness and the approximation of operators, Cambridge Tracts in Mathematics, vol. 98, Cambridge University Press, Cambridge, 1990. MR 92e:47002
  • 5. R. M. Dudley, Universal Donsker classes and metric entropy, Ann. Probab. 15 (1987), no. 4, 1306-1326. MR 88g:60081
  • 6. Fuchang Gao, Metric entropy of convex hulls, Israel J. Math. 123 (2001), 359-364. MR 2002c:46044
  • 7. -, Metric entropy of absolute convex hulls in Hilbert spaces, Bull. London Math. Soc. 36 (2004), no. 4, 460-468.
  • 8. Siegfried Graf and Harald Luschgy, Foundations of quantization for probability distributions, Lecture Notes in Mathematics, vol. 1730, Springer-Verlag, Berlin, 2000. MR 2001m:60043
  • 9. Atsuyuki Okabe, Barry Boots, and Kokichi Sugihara, Spatial tessellations: concepts and applications of Vorono{\u{\i}}\kern.15em diagrams, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, John Wiley & Sons Ltd., Chichester, 1992, With a foreword by D. G. Kendall. MR 94a:52033

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Additional Information

Fuchang Gao
Affiliation: Department of Mathematics, University of Idaho, Moscow, Idaho 83843

Keywords: Leibniz half-space, Voronoi region, metric entropy
Received by editor(s): August 26, 2003
Received by editor(s) in revised form: February 17, 2004
Published electronically: November 19, 2004
Additional Notes: This research was partially supported by NSF grant EPS-0132626 and a seed grant from the University of Idaho.
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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