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
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 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.
 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
(2004k:28001)
 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 (88j:52001)
 3.
Hallard
T. Croft, Kenneth
J. Falconer, and Richard
K. Guy, Unsolved problems in geometry, Problem Books in
Mathematics, SpringerVerlag, New York, 1991. Unsolved Problems in
Intuitive Mathematics, II. MR 1107516
(92c:52001)
 4.
E.
Egerváry, On the smallest convex cover of a simple arc of
spacecurve, Publ. Math. Debrecen 1 (1949),
65–70. MR
0036021 (12,46e)
 5.
Herbert
Federer, Geometric measure theory, Die Grundlehren der
mathematischen Wissenschaften, Band 153, SpringerVerlag New York Inc., New
York, 1969. MR
0257325 (41 #1976)
 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
(2008f:52014)
 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
(56 #16284)
 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
(22 #7058), http://dx.doi.org/10.1090/S00029939196001162630
 9.
Z.
A. Melzak, Numerical evaluation of an
isoperimetric constant, Math. Comp. 22 (1968), 188–190. MR 0223976
(36 #7023), http://dx.doi.org/10.1090/S00255718196802239764
 10.
P.
A. P. Moran, On a problem of S. Ulam, J. London Math. Soc.
21 (1946), 175–179. MR 0020799
(8,597n)
 11.
Frank
Morgan, (𝑀,𝜀,𝛿)minimal
curve regularity, Proc. Amer. Math. Soc.
120 (1994), no. 3,
677–686. MR 1169884
(94e:49018), http://dx.doi.org/10.1090/S00029939199411698843
 12.
A.
A. Nudel′man, Isoperimetric problems for the convex hulls of
polygonal lines and curves in higherdimensional spaces, Mat. Sb.
(N.S.) 96(138) (1975), 294–313, 344 (Russian). MR 0375090
(51 #11286)
 13.
I.
J. Schoenberg, An isoperimetric inequality for closed curves convex
in evendimensional Euclidean spaces, Acta Math. 91
(1954), 143–164. MR 0065944
(16,508b)
 14.
Alan
Siegel, Some Didotype inequalities, Elem. Math.
56 (2001), no. 1, 17–20. MR
1818263, http://dx.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
(2002m:52009), http://dx.doi.org/10.1007/s0045400100636
 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 (97c:58028)
 1.
 L. Ambrosio, P. Tilli,
Topics on analysis in metric spaces. Oxford University Press, Oxford, 2004. MR 2039660 (2004k:28001)
 2.
 T. Bonnesen, W. Fenchel, Theory of convex bodies, translated from the German and edited by L. Boron, C. Christenson and B. Smith. BCS Assoc., Moscow, 1987. MR 920366 (88j:52001)
 3.
 H. T. Croft, K. J. Falconer, R.K. Guy, Unsolved problems in geometry. Problem Books in Mathematics. Unsolved Problems in Intuitive Mathematics, II. SpringerVerlag, New York, 1991. MR 1107516 (92c:52001)
 4.
 E. Egerváry, On the smallest convex cover of a simple arc of spacecurve. Publ. Math. Debrecen 1 (1949), 6570. MR 0036021 (12:46e)
 5.
 H. Federer, Geometric measure theory. SpringerVerlag, New York, 1969. MR 0257325 (41:1976)
 6.
 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
 7.
 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)
 8.
 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)
 9.
 Z. A. Melzak, Numerical evaluation of an isoperimetric constant. Math. Comp. 22 (1968), 188190. MR 0223976 (36:7023)
 10.
 P. A. P. Moran, On a problem of S. Ulam. J. London Math. Soc. 21 (1946), 175179. MR 0020799 (8:597n)
 11.
 F. Morgan, (M,e,d)minimal curve regularity. Proc. Amer. Math. Soc. 120 (1994), 677686. MR 1169884 (94e:49018)
 12.
 A. A. Nudel'man,
Isoperimetric problems for the convex hulls of polygonal lines and curves in multidimensional spaces. (Russian) Mat. Sb. (N.S.) 96(138) (1975), 294313. MR 0375090 (51:11286)
 13.
 I. J. Schoenberg, An isoperimetric inequality for closed curves convex in evendimensional Euclidean spaces. Acta Math. 91 (1954), 143164. MR 0065944 (16:508b)
 14.
 A. Siegel, Some Didotype inequalities. Elem. Math. 56 (2001), no. 1, 1720. MR 1818263
 15.
 A. Siegel, A Dido problem as modernized by Fejes Tóth. Discrete Comput. Geom. 27 (2002), no. 2, 227238. MR 1880939 (2002m:52009)
 16.
 V. A. Zalgaller, Extremal problems on the convex hull of a space curve. (Russian) Algebra i Analiz 8 (1996), no. 3, 113. Translation in St. Petersburg Math. J. 8 (1997), no. 3, 369379. MR 1402285 (97c:58028)
Similar Articles
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:
http://dx.doi.org/10.1090/S0002994710047343
PII:
S 00029947(10)047343
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.
