Reduced boundaries and convexity
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- by David G. Caraballo PDF
- Proc. Amer. Math. Soc. 141 (2013), 1775-1782 Request permission
Abstract:
We establish strong, new connections between convex sets and geometric measure theory. We use geometric measure theory to improve several standard theorems from the theory of convex sets, which have found wide application in fields such as functional analysis, economics, optimization, and control theory. For example, we prove that a closed subset $K$ of $\mathbb {R}^{n}$ with non-empty interior is convex if and only if it has locally finite perimeter in $\mathbb {R}^{n}$ and has a supporting hyperplane through each point of its reduced boundary. This refines the standard result that such a set $K$ is convex if and only if it has a supporting hyperplane through each point of its topological boundary, which may be much larger than the reduced boundary. Thus, the reduced boundary from geometric measure theory contains all the convexity information for such a set $K$. We similarly refine a standard separation theorem, as well as a representation theorem for convex sets. We then extend all of our results to other notions of boundary from the literature and deduce the corresponding classical results from convex analysis as special cases.References
- Luigi Ambrosio, Nicola Fusco, and Diego Pallara, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000. MR 1857292
- David G. Caraballo, Crystals and polycrystals in $\Bbb R^n$: lower semicontinuity and existence, J. Geom. Anal. 18 (2008), no. 1, 68–88. MR 2365668, DOI 10.1007/s12220-007-9006-7
- David G. Caraballo, Convexity, local simplicity, and reduced boundaries of sets, J. Convex Anal. 18 (2011), no. 3, 823–832. MR 2858096
- D. G. Caraballo, On Tietze’s convexity theorem and other local criteria for convexity, in preparation.
- Ennio De Giorgi, Su una teoria generale della misura $(r-1)$-dimensionale in uno spazio ad $r$ dimensioni, Ann. Mat. Pura Appl. (4) 36 (1954), 191–213 (Italian). MR 62214, DOI 10.1007/BF02412838
- Ennio De Giorgi, Nuovi teoremi relativi alle misure $(r-1)$-dimensionali in uno spazio ad $r$ dimensioni, Ricerche Mat. 4 (1955), 95–113 (Italian). MR 74499
- Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR 1158660
- Herbert Federer, The Gauss-Green theorem, Trans. Amer. Math. Soc. 58 (1945), 44–76. MR 13786, DOI 10.1090/S0002-9947-1945-0013786-6
- Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
- Enrico Giusti, Minimal surfaces and functions of bounded variation, Monographs in Mathematics, vol. 80, Birkhäuser Verlag, Basel, 1984. MR 775682, DOI 10.1007/978-1-4684-9486-0
- Steven G. Krantz and Harold R. Parks, The geometry of domains in space, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Boston, Inc., Boston, MA, 1999. MR 1730695, DOI 10.1007/978-1-4612-1574-5
- Steven R. Lay, Convex sets and their applications, Robert E. Krieger Publishing Co., Inc., Malabar, FL, 1992. Revised reprint of the 1982 original. MR 1170565
- Pertti Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995. Fractals and rectifiability. MR 1333890, DOI 10.1017/CBO9780511623813
- Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1993. MR 1216521, DOI 10.1017/CBO9780511526282
- Roger Webster, Convexity, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1994. MR 1443208
- Martin L. Weitzman, An ‘economics proof’ of the supporting hyperplane theorem, Econom. Lett. 68 (2000), no. 1, 1–6. MR 1765148, DOI 10.1016/S0165-1765(00)00227-5
Additional Information
- David G. Caraballo
- Affiliation: Department of Mathematics and Statistics, St. Mary’s Hall, 3rd floor, Georgetown University, Washington, DC 20057-1233
- MR Author ID: 769888
- Received by editor(s): October 26, 2010
- Received by editor(s) in revised form: February 11, 2011
- Published electronically: January 29, 2013
- Communicated by: Tatiano Toro
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 141 (2013), 1775-1782
- MSC (2010): Primary 52A20, 52A30, 28A75
- DOI: https://doi.org/10.1090/S0002-9939-2013-11099-3
- MathSciNet review: 3020862