Isoperimetric inequalities for convex hulls and related questions

Author:
Paolo Tilli

Journal:
Trans. Amer. Math. Soc. **362** (2010), 4497-4509

MSC (2010):
Primary 52A10, 52B60, 52A40

DOI:
https://doi.org/10.1090/S0002-9947-10-04734-3

Published electronically:
April 5, 2010

MathSciNet review:
2645038

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the problem of maximizing the Lebesgue measure of the convex hull of a connected compact set of prescribed one-dimensional 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.

**1.**Luigi Ambrosio and Paolo Tilli,*Topics on analysis in metric spaces*, Oxford Lecture Series in Mathematics and its Applications, vol. 25, Oxford University Press, Oxford, 2004. MR**2039660****2.**T. Bonnesen and W. Fenchel,*Theory of convex bodies*, BCS Associates, Moscow, ID, 1987. Translated from the German and edited by L. Boron, C. Christenson and B. Smith. MR**920366****3.**Hallard T. Croft, Kenneth J. Falconer, and Richard K. Guy,*Unsolved problems in geometry*, Problem Books in Mathematics, Springer-Verlag, New York, 1991. Unsolved Problems in Intuitive Mathematics, II. MR**1107516****4.**E. Egerváry,*On the smallest convex cover of a simple arc of space-curve*, Publ. Math. Debrecen**1**(1949), 65–70. MR**0036021****5.**Herbert Federer,*Geometric measure theory*, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR**0257325****6.**Michele Gori,*On a maximization problem for the convex hull of connected systems of segments*, J. Convex Anal.**14**(2007), no. 1, 49–68. MR**2310428****7.**M. G. Kreĭn and A. A. Nudel′man,*The Markov moment problem and extremal problems*, American Mathematical Society, Providence, R.I., 1977. Ideas and problems of P. L. Čebyšev and A. A. Markov and their further development; Translated from the Russian by D. Louvish; Translations of Mathematical Monographs, Vol. 50. MR**0458081****8.**Z. A. Melzak,*The isoperimetric problem of the convex hull of a closed space curve.*, Proc. Amer. Math. Soc.**11**(1960), 265–274. MR**0116263**, https://doi.org/10.1090/S0002-9939-1960-0116263-0**9.**Z. A. Melzak,*Numerical evaluation of an isoperimetric constant*, Math. Comp.**22**(1968), 188–190. MR**0223976**, https://doi.org/10.1090/S0025-5718-1968-0223976-4**10.**P. A. P. Moran,*On a problem of S. Ulam*, J. London Math. Soc.**21**(1946), 175–179. MR**0020799**, https://doi.org/10.1112/jlms/s1-21.3.175**11.**Frank Morgan,*(𝑀,𝜀,𝛿)-minimal curve regularity*, Proc. Amer. Math. Soc.**120**(1994), no. 3, 677–686. MR**1169884**, https://doi.org/10.1090/S0002-9939-1994-1169884-3**12.**A. A. Nudel′man,*Isoperimetric problems for the convex hulls of polygonal lines and curves in higher-dimensional spaces*, Mat. Sb. (N.S.)**96(138)**(1975), 294–313, 344 (Russian). MR**0375090****13.**I. J. Schoenberg,*An isoperimetric inequality for closed curves convex in even-dimensional Euclidean spaces*, Acta Math.**91**(1954), 143–164. MR**0065944**, https://doi.org/10.1007/BF02393429**14.**Alan Siegel,*Some Dido-type inequalities*, Elem. Math.**56**(2001), no. 1, 17–20. MR**1818263**, https://doi.org/10.1007/s000170050085**15.**A. Siegel,*A Dido problem as modernized by Fejes Tóth*, Discrete Comput. Geom.**27**(2002), no. 2, 227–238. MR**1880939**, https://doi.org/10.1007/s00454-001-0063-6**16.**V. A. Zalgaller,*Extremal problems on the convex hull of a space curve*, Algebra i Analiz**8**(1996), no. 3, 1–13 (Russian, with Russian summary); English transl., St. Petersburg Math. J.**8**(1997), no. 3, 369–379. MR**1402285**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2010):
52A10,
52B60,
52A40

Retrieve articles in all journals with MSC (2010): 52A10, 52B60, 52A40

Additional Information

**Paolo Tilli**

Affiliation:
Dipartimento di Matematica, Politecnico di Torino, 10129 Torino, Italy

Email:
paolo.tilli@polito.it

DOI:
https://doi.org/10.1090/S0002-9947-10-04734-3

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.