Abstract

It is well known that any measure in S2S^2 satisfying certain simple conditions is the surface measure of a bounded convex body in R3\mathbb{R}^3. It is also known that a local perturbation of the surface measure may lead to a nonlocal perturbation of the corresponding convex body. We prove that, under mild conditions on the convex body, there are families of perturbations of its surface measure forming line segments, with the original measure at the midpoint, corresponding to local perturbations of the body. Moreover, there is, in a sense, a huge amount of such families. We apply this result to Newton's problem of minimal resistance for convex bodies.

Author information

Contact details are reproduced from the original publication and may be historical.

Alexander Plakhov

Center for R&D in Mathematics and Applications, Dept. of Mathematics, University of Aveiro, Portugal
and: Institute for Information Transmission Problems, Moscow, Russia

plakhov@ua.pt

Keywords

  • Convex sets
  • Blaschke addition
  • Newton's problems of minimal resistance

Mathematics Subject Classification

52A15, 52A40, 49Q10, 49Q20

References

  1. A. D. Alexandrov: Selected Works. I: Selected Scientific Papers, Chapter V, §3, Yu. G. Reshetnyak and S. S. Kutateladze (eds.), Gordon & Breach, Philadelphia et al. (1996).Google Scholar
  2. V. Alexandrov, N. Kopteva, S. S. Kutateladze, Blaschke addition and convex polyhedra, arXiv: 0502345 (2005).Google Scholar
  3. F. Brock, V. Ferone, B. Kawohl: A symmetry problem in the calculus of variations, Calc. Var. 4 (1996) 593–599.https://doi.org/10.1007/BF01261764Google Scholar
  4. G. Buttazzo, V. Ferone, B. Kawohl: Minimum problems over sets of concave functions and related questions, Math. Nachrichten 173 (1995) 71–89.https://doi.org/10.1002/mana.19951730106Google Scholar
  5. G. Buttazzo, P. Guasoni: Shape optimization problems over classes of convex domains, J. Convex Analysis 4(2) (1997) 343–351.https://doi.org/10.68381/jca04019Google Scholar
  6. G. Buttazzo, B. Kawohl: On Newton’s problem of minimal resistance, Math. Intell. 15 (1993) 7–12.https://doi.org/10.1007/BF03024318Google Scholar
  7. G. Carlier, T. Lachand-Robert: Convex bodies of optimal shape, J. Convex Analysis 10(1) (2003) 265–273.https://doi.org/10.68381/jca1015Google Scholar
  8. M. Comte, T. Lachand-Robert: Newton’s problem of the body of minimal resistance under a single-impact assumption, Calc. Var. Partial Differ. Equ. 12 (2001) 173–211.https://doi.org/10.1007/PL00009911Google Scholar
  9. M. Comte, T. Lachand-Robert: Existence of minimizers for Newton’s problem of the body of minimal resistance under a single-impact assumption, J. Anal. Math. 83 (2001) 313–335.https://doi.org/10.1007/BF02790266Google Scholar
  10. T. Lachand-Robert, E. Oudet: Minimizing within convex bodies using a convex hull method, SIAM J. Optim. 16 (2006) 368–379.https://doi.org/10.1137/040608039Google Scholar
  11. T. Lachand-Robert, M. A. Peletier: Newton’s problem of the body of minimal resistance in the class of convex developable functions, Math. Nachrichten 226 (2001) 153–176.https://doi.org/10.1002/1522-2616(200106)226:1<153::AID-MANA153>3.0.CO;2-2Google Scholar
  12. T. Lachand-Robert, M. A. Peletier: An example of non-convex minimization and an application to Newton’s problem of the body of least resistance, Ann. Inst. H. Poincaré, Anal. Non Lin. 18 (2001) 179–198.https://doi.org/10.1016/s0294-1449(00)00062-7Google Scholar
  13. A. Plakhov: Exterior Billiards. Systems with Impacts Outside Bounded Domains, Springer, New York (2012).https://doi.org/10.1007/978-1-4614-4481-7Google Scholar
  14. A. Plakhov, D. Torres: Newton’s aerodynamic problem in media of chaotically moving particles, Sbornik: Math. 196 (2005) 885–933.https://doi.org/10.1070/SM2005v196n06ABEH000904Google Scholar
  15. G. Wachsmuth: The numerical solution of Newton’s problem of least resistance, Math. Program. A 147 (2014) 331–350.https://doi.org/10.1007/s10107-014-0756-2Google Scholar
  16. H. Zouaki: Convex set symmetry measurement using Blaschke addition, Pattern Recognition 36 (2003) 753–763.https://doi.org/10.1016/S0031-3203(02)00090-0Google Scholar

Suggested citation

A. Plakhov. “Local Properties of the Surface Measure of Convex Bodies.” Journal of Convex Analysis 26 (2019), No. 4, 1373–1402. https://doi.org/10.68381/jca26073

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2019. Rights now held by Banach Press.