Isoperimetric inequalities for convex hulls and related questions
Author:
Paolo Tilli
Journal:
Trans. Amer. Math. Soc. 362 (2010), 44974509
MSC (2010):
Primary 52A10, 52B60, 52A40
Published electronically:
April 5, 2010
MathSciNet review:
2645038
Fulltext PDF Free Access
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Abstract: We consider the problem of maximizing the Lebesgue measure of the convex hull of a connected compact set of prescribed onedimensional Hausdorff measure. In dimension two, we prove that the only solutions are semicircles. In higher dimensions, we prove some isoperimetric inequalities for convex hulls of connected sets; we focus on a classical open problem and discuss a possible new approach.
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 E. Egerváry, On the smallest convex cover of a simple arc of spacecurve. Publ. Math. Debrecen 1 (1949), 6570. MR 0036021 (12:46e)
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 H. Federer, Geometric measure theory. SpringerVerlag, New York, 1969. MR 0257325 (41:1976)
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 M. Gori, On a maximization problem for the convex hull of connected systems of segments. Journal of Convex Analysis 14 (2007), no. 1, 4968. MR 2310428
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 M. G. Kreĭn, A.A. Nudel'man, The Markov moment problem and extremal problems. Amer. Math. Soc., Providence, R.I., 1977. MR 0458081 (56:16284)
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 Z. A. Melzak, The isoperimetric problem of the convex hull of a closed space curve. Proc. Amer. Math. Soc. 11 (1960), 265274. MR 0116263 (22:7058)
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Additional Information
Paolo Tilli
Affiliation:
Dipartimento di Matematica, Politecnico di Torino, 10129 Torino, Italy
Email:
paolo.tilli@polito.it
DOI:
http://dx.doi.org/10.1090/S0002994710047343
Received by editor(s):
November 22, 2006
Received by editor(s) in revised form:
January 8, 2008
Published electronically:
April 5, 2010
Article copyright:
© Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
