Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-26T22:03:07.670Z Has data issue: false hasContentIssue false
  • English
  • Français

Shadow boundaries of typical convex bodies. measure properties

Part of: General convexity

Published online by Cambridge University Press:  26 February 2010

Peter M. Gruber  and
Hartwig Sorger
Peter M. Gruber
Affiliation:
Abteilung fr Analysis, Technische Universitt Wien, Wiedner Hauptstrae 8-101142, A-1040 Vienna, Austria.
Hartwig Sorger
Affiliation:
Abteilung fr Analysis, Technische Universitt Wien, Wiedner Hauptstrajse 8-101142, A-1040 Vienna, Austria.
Get access

Abstract

For a typical convex body in Ed a typical shadow boundary under parallel illumination has infinite (d - 2)-dimensional Hausdorff measurewhile having Hausdorff dimension d 2.

MSC classification

Secondary: 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces)
Type
Research Article
Information
Mathematika , Volume 36 , Issue 1 , June 1989 , pp. 142 - 152
Copyright
Copyright University College London 1989

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Blaschke, W.. Rumliche Variationsprobleme mit symmetrischer Transversalittsbedingung. Ber. Verh. Schs. Akad. Wiss. Math.-naturw. KL, 68 (1916), 5055.Google Scholar
2.Blaschke, W.. Kreis und Kugel (Gschen, Leipzig, 1916, Chelsea, New York, 1949, de Gruyter, Berlin, 1956).Google Scholar
3.Busemann, H.. Convex Surfaces (Interscience, New York, 1958).Google Scholar
4.Falconer, K. J.. The Geometry of Fractal Sets (Cambridge Univ. Press, Cambridge, 1985).CrossRefGoogle Scholar
5.Federer, H.. The (, k) rectifiable subsets of n space. Trans. Amer. Math. Soc., 62 (1947), 114192.Google Scholar
6.Federer, H.. Measure and area. Bull. Amer. Math. Soc., 58 (1952), 306378.CrossRefGoogle Scholar
7.Gruber, P. M.. Die meisten konvexen Krper sind glatt, aber nicht zu glatt. Math. Ann., 229 (1977), 259266.Google Scholar
9.Gruber, P. M.. In most cases approximation is irregular. Rend. Sem. Mat. Univ. Politecn. Torino, 41 (1983), 1833.Google Scholar
9.Gruber, P. M.. Aspects of convexity and its applications. Expos. Math., 2 (1984), 4783.Google Scholar
10.Gruber, P. M.. Results of Baire category type in convexity. Ann. New York Acad. Sci., 440 (1985), 163169.CrossRefGoogle Scholar
11.Gruber, P. M.. The size of typical compact sets, continua and curves. Submitted for publication.Google Scholar
12.Gruber, P. M. and Hbinger, J.. Kennzeichnungen von Ellipsoiden mit Anwendungen. In:Jahrb. berblicke Math., 1976, 929.Google Scholar
13.Holmes, R. G.. Geometrical Functional Analysis and its Applications (Springer, New York, 1976).Google Scholar
14.Kieselman, C. O.. How smooth is the shadow of a smooth convex body? J. London Math. Soc. (2), 33 (1986), 101109.CrossRefGoogle Scholar
15.Klee, V.. Some new results on smoothness and rotundity in normed linear spaces. Math. Ann., 139 (1959), 5163.CrossRefGoogle Scholar
16.Kolmogorov, A. N. and Tihomirov, V. M.. -entropy and -capacity. Uspekhi Mat. Nauk, 14 (1959), 386 = Amer. Math. Soc. Transl. (2), 17 (1961), 277364.Google Scholar
17.Laugwitz, D.. Differentialgeometrie in Vektorrumen (Vieweg, Braunschweig, 1965).Google Scholar
18.Leichtweiss, K.. Konvexe Mengen (Springer, Berlin, 1980).CrossRefGoogle Scholar
19.Oxtoby, J. C.. Measure and Category (Springer, New York, 1971).Google Scholar
20.Rogers, C. A.. Hausdorff Measures (Cambridge Univ. Press, Cambridge, 1970).Google Scholar
21.Steenaerts, P.. Mittlere Schattengrenzenlnge konvexer Krper. Results in Math., 8 (1985), 5477.CrossRefGoogle Scholar
22.Zamfirescu, T.. Using Baire categories in geometry. Rend. Sem. Mat. Univ. Politecn. Torino, 43 (1985), 6788.Google Scholar
23.Zamfirescu, T.. Too long shadow boundaries. Proc. Amer. Math. Soc., 103 (1988), 587590.CrossRefGoogle Scholar