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Transactions of the American Mathematical Society

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Volume bounds for shadow covering


Authors: Christina Chen, Tanya Khovanova and Daniel A. Klain
Journal: Trans. Amer. Math. Soc. 366 (2014), 1161-1177
MSC (2010): Primary 52A20
DOI: https://doi.org/10.1090/S0002-9947-2013-05855-2
Published electronically: August 20, 2013
MathSciNet review: 3145726
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Abstract: For $ n \geq 2$ a construction is given for a large family of compact convex sets $ K$ and $ L$ in $ \mathbb{R}^n$ such that the orthogonal projection $ L_u$ onto the subspace $ u^\perp $ contains a translate of the corresponding projection $ K_u$ for every direction $ u$, while the volumes of $ K$ and $ L$ satisfy $ V_n(K) > V_n(L).$

It is subsequently shown that if the orthogonal projection $ L_u$ onto the subspace $ u^\perp $ contains a translate of $ K_u$ for every direction $ u$, then the set $ \frac {n}{n-1} L$ contains a translate of $ K$. It follows that

$\displaystyle V_n(K) \leq \left (\frac {n}{n-1}\right )^n V_n(L).$

In particular, we derive a universal constant bound

$\displaystyle V_n(K) \leq 2.942 \, V_n(L),$

independent of the dimension $ n$ of the ambient space. Related results are obtained for projections onto subspaces of some fixed intermediate co-dimension. Open questions and conjectures are also posed.

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  • 1. K. Ball, Some remarks on the geometry of convex sets, Geometric Aspects of Functional Analysis (J. Lindenstrauss and V.D. Milman, eds.), Springer Lecture Notes in Mathematics, vol. 1317, Springer Verlag, Berlin, 1988. MR 950983 (89h:52009)
  • 2. K. Ball, Shadows of convex bodies, Trans. Amer. Math. Soc. 327 (1991), 891-901. MR 1035998 (92a:52011)
  • 3. T. Bonnesen and W. Fenchel, Theory of Convex Bodies, BCS Associates, Moscow, Idaho, 1987. MR 920366 (88j:52001)
  • 4. J. Bourgain, On the Busemann-Petty problem for perturbations of the ball, Geom. Funct. Anal. 1 (1991), 1-13. MR 1091609 (92c:52008)
  • 5. H. Busemann and C. M. Petty, Problems on convex bodies, Math. Scand. 4 (1956), 88-94. MR 0084791 (18:922b)
  • 6. G. D. Chakerian, Inequalities for the difference body of a convex body, Proc. Amer. Math. Soc. 18 (1967), 879-884. MR 0218972 (36:2055)
  • 7. C. Chen, Maximizing volume ratios for shadow covering by tetrahedra, arXiv:1201.2580v1 (2012).
  • 8. R. J. Gardner, Intersection bodies and the Busemann-Petty problem, Trans. Amer. Math. Soc. 342 (1994), 435-445. MR 1201126 (94e:52008)
  • 9. R. J. Gardner, A positive answer to the Busemann-Petty problem in three dimensions, Ann. Math. (2) 140 (1994), 435-447. MR 1298719 (95i:52005)
  • 10. R. J. Gardner, The Brunn-Minkowski inequality, Bull. Amer. Math. Soc. 39 (2002), 355-405. MR 1898210 (2003f:26035)
  • 11. R. J. Gardner, Geometric Tomography (2nd Ed.), Cambridge University Press, New York, 2006. MR 2251886 (2007i:52010)
  • 12. R. J. Gardner, A. Koldobsky, and T. Schlumprecht, An analytic solution to the Busemann-Petty problem on sections of convex bodies, Ann. Math. (2) 149 (1999), 691-703. MR 1689343 (2001b:52011)
  • 13. R. J. Gardner and A. Volčič, Convex bodies with similar projections, Proc. Amer. Math. Soc. 121 (1994), 563-568. MR 1185262 (94h:52005)
  • 14. A. Giannopoulos, A note on a problem of H. Busemann and C. M. Petty concerning sections of symmetric convex bodies, Mathematika 37 (1990), 239-244. MR 1099772 (92c:52009)
  • 15. H. Groemer, Ein Satz über konvexe Körper und deren Projektionen, Portugal. Math. 21 (1962), 41-43. MR 0142055 (25:5449)
  • 16. H. Hadwiger, Gegenseitige Bedeckbarkeit zweier Eibereiche und Isoperimetrie, Vierteljschr. Naturforsch. Gesellsch. Zürich 86 (1941), 152-156. MR 0007274 (4:112c)
  • 17. H. Hadwiger, Überdeckung ebener Bereiche durch Kreise und Quadrate, Comment. Math. Helv. 13 (1941), 195-200. MR 0004995 (3:90f)
  • 18. H. Hadwiger, Seitenrisse konvexer Körper und Homothetie, Elem. Math. 18 (1963), 97-98. MR 0155232 (27:5168)
  • 19. D. Klain, Containment and inscribed simplices, Indiana Univ. Math. J. 59 (2010), 1231-1244. MR 2815032
  • 20. D. Klain, Covering shadows with a smaller volume, Adv. Math. 224 (2010), 601-619. MR 2609017 (2011d:52012)
  • 21. D. Klain, If you can hide behind it, can you hide inside it?, Trans. Amer. Math. Soc. 363 (2011), 4585-4601. MR 2806685
  • 22. D. Klain and G.-C. Rota, Introduction to Geometric Probability, Cambridge University Press, New York, 1997. MR 1608265 (2001f:52009)
  • 23. A. Koldobsky, Intersection bodies, positive definite distributions, and the Busemann-Petty problem, Amer. J. Math. 120 (1998), 827-840. MR 1637955 (99i:52005)
  • 24. D. G. Larman and C. A. Rogers, The existence of a centrally symmetric convex body with central sections that are unexpectedly small, Mathematika 22 (1975), 164-175. MR 0390914 (52:11737)
  • 25. E. Lutwak, Intersection bodies and dual mixed volumes, Adv. Math. 71 (1988), 232-261. MR 963487 (90a:52023)
  • 26. E. Lutwak, Containment and circumscribing simplices, Discrete Comput. Geom. 19 (1998), 229-235. MR 1600051 (99a:52009)
  • 27. A. Papadimitrakis, On the Busemann-Petty problem about convex centrally symmetric bodies in $ \mathbb{R}^n$, Mathematika 39 (1992), 258-266. MR 1203282 (94a:52019)
  • 28. C. M. Petty, Projection bodies, Proceedings, Coll. Convexity, Copenhagen, 1965, vol. Københavns Univ. Mat. Inst., 1967, pp. 234-241. MR 0216369 (35:7203)
  • 29. C. A. Rogers, Sections and projections of convex bodies, Portugal. Math. 24 (1965), 99-103. MR 0198344 (33:6502)
  • 30. C. A. Rogers and G. C. Shephard, The difference body of a convex body, Arch. Math. 8 (1957), 220-233. MR 0092172 (19:1073f)
  • 31. L. A. Santaló, Integral Geometry and Geometric Probability, Addison-Wesley, Reading, MA, 1976. MR 0433364 (55:6340)
  • 32. R. Schneider, Zur einem Problem von Shephard über die Projektionen konvexer Körper., Math Z. 101 (1967), 71-82. MR 0218976 (36:2059)
  • 33. R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Cambridge University Press, New York, 1993. MR 1216521 (94d:52007)
  • 34. G. C. Shephard, Shadow systems of convex bodies, Israel J. Math. 2 (1964), 229-236. MR 0179686 (31:3931)
  • 35. V. Soltan, Convex sets with homothetic projections, Beiträge Algebra Geom. 51 (2010), 237-249. MR 2650489 (2011c:52010)
  • 36. R. Webster, Convexity, Oxford University Press, New York, 1994. MR 1443208 (98h:52001)
  • 37. G. Zhang, Intersection bodies and the Busemann-Petty inequalities in $ \mathbb{R}^4$, Ann. Math. (2) 140 (1994), no. 2, 331-346. MR 1298716 (95i:52004)
  • 38. G. Zhang, A positive solution to the Busemann-Petty problem in $ \mathbb{R}^4$, Ann. Math. (2) 149 (1999), 535-543. MR 1689339 (2001b:52010)
  • 39. J. Zhou, The sufficient condition for a convex body to contain another in $ \mathbb{R}^4$, Proc. Amer. Math. Soc. 121 (1994), 907-913. MR 1184090 (94i:52007)
  • 40. J. Zhou, Sufficient conditions for one domain to contain another in a space of constant curvature, Proc. Amer. Math. Soc. 126 (1998), 2797-2803. MR 1451838 (98k:52016)

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Additional Information

Christina Chen
Affiliation: Newton North High School, Newton, Massachusetts 02460

Tanya Khovanova
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139

Daniel A. Klain
Affiliation: Department of Mathematical Sciences, University of Massachusetts Lowell, Lowell, Massachusetts 01854
Email: Daniel\textunderscore{}Klain@uml.edu

DOI: https://doi.org/10.1090/S0002-9947-2013-05855-2
Received by editor(s): October 22, 2011
Published electronically: August 20, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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