Abstract
Ifv r is ther-dimensional volume of ther-simplex formed byr+1 points taken at random from a compact setK in ℝn, withr≤n, andh is a (strictly) increasing function, then the (unique) compact set that gives the minimum expected value ofh o v r, is proved to be the ellipsoid (whenr=n) and the ball (whenr<n) almost everywhere. This result is established by using a single integral inequality for centrally symmetric quasiconvex functions integrated over compact rectangles.
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References
Anderson, T. W. (1955). The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities.Proc. Am. Math. Soc. 6, 170–176.
Berge, C. (1963).Topological Spaces, Oliver and Boyd, Edinburgh.
Blaschke, W. (1917). Über affine Geometrie XI: Lösung des “Vierpunktproblems” von Sylvester aus der Theorie der geometrischen Wahrscheinlichkeiten.Leipziger Ber. 69, 436–453.
Blaschke, W. (1918). Eine Isoperimetrische Eigenschaft des Kreises.Math. Z. 1, 52–57.
Blaschke, W. (1923).Vorlesungen über Differentialgeometrie II: Affine Differentialgeometrie, Springer-Verlag, Berlin.
Borden, R. S. (1983).A Course in Advanced Calculus, North-Holland, New York.
Buchta, C. (1980). Zufällige Polyeder—Eine Übersicht. InZahlentheoretische Analysis (ed. by Hlawka, E. Ed.), Lecture Notes in Mathematics 1114, Springer-Verlag, Berlin.
Carleman, T. (1919). Über eine isoperimetrisch Aufgabe and ihre physikalischen anwendungen.Math. Z. 3, 1–7.
Cormier, R. J. (1971). Steiner Symmetrization inE n.Rev. Mat. Hisp-Amer. 31(4), 197–204.
Groemer, H. (1973). On some mean values associated with a randomly selected simplex in a convex set.Pacific J. Math. 45, 525–533.
Groemer, H. (1982). On the average size of polytopes in a convex set.Geom. Ded. 13, 47–62.
Hadwiger, H. (1957).Vorlesungen über Inhalt, Oberfläche und Isoperimetrie, Springer-Verlag, Berlin.
Henze, N. (1983). Random triangles in convex regions.J. Appl. Prob. 20, 111–125.
Pfiefer, R. (1982). The extrema of geometric mean values, Dissertation, University of California, Davis.
Pólya, G., and Szegö, G. (1951).Isoperimetric Inequalities in Mathematical Physics, Princeton University Press, Princeton, New Jersey.
Schneider, R. (1988). Random approximation of convex sets.J. Microscopy 151, 211–227.
Schöpf, P. (1977). Gewichtete Volumsmittelwerte von Simplices, welche zufällig in einem konvexen Körper des ℝn gewahlt werden.Monatsh. Math. 83, 331–337.
Williamson, J. H. (1962).Lebesgue Integration, Holt, Rinehart and Winston, New York.
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Pfiefer, R.E. Maximum and minimum sets for some geometric mean values. J Theor Probab 3, 169–179 (1990). https://doi.org/10.1007/BF01045156
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DOI: https://doi.org/10.1007/BF01045156