Math 2.0

Convex and discrete geometry Convex geometric analysis and geometric inequalities Problem

Busemann–Petty problem

Builds on Convex bodyLebesgue measure

Problem

Let n≥2n \ge 2 and let K,L⊆ℝnK, L \subseteq \R^n be compact convex bodies with nonempty interior, both symmetric about the origin. Assume that for every hyperplane HH through the origin, vol⁡n−1(K∩H)≤vol⁡n−1(L∩H),\operatorname{vol}_{n-1}(K \cap H) \le \operatorname{vol}_{n-1}(L \cap H), where vol⁡n−1\operatorname{vol}_{n-1} is the (n−1)(n-1)-dimensional Lebesgue measure on HH. Does it follow that vol⁡n(K)≤vol⁡n(L)\operatorname{vol}_n(K) \le \operatorname{vol}_n(L)?

Remarks

The problem is solved: the answer is yes for n≤4n \le 4 and no for n≥5n \ge 5. For n=2n = 2 the answer is yes: a chord through the center in direction uu has length 2ρ(u)2\rho(u), where ρ\rho is the radial function, so the hypothesis gives K⊆LK \subseteq L. Lutwak showed that the answer in ℝn\R^n is yes if and only if every origin-symmetric convex body in ℝn\R^n is an intersection body. Counterexamples in high dimensions were found first by Larman and Rogers, and the remaining dimensions were settled through work of Ball, Bourgain, Giannopoulos, Papadimitrakis, Gardner and Zhang, among others. A unified Fourier-analytic solution in all dimensions was given by Gardner, Koldobsky and Schlumprecht.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

Show the path as a graph
Borel sigma-algebra, concept, no evidence levelBorel sigma-algebraconceptConvex body, concept, no evidence levelConvex bodyconceptEuclidean space, object, no evidence levelEuclidean spaceobjectOuter measure, concept, no evidence levelOuter measureconceptLebesgue measure, object, no evidence levelLebesgue measureobjectBusemann–Petty problem, problem, no evidence levelBusemann–Petty problemproblem
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Borel sigma-algebra concept
  2. Convex body concept
  3. Euclidean space object
  4. Outer measure concept
  5. Lebesgue measure object
  6. Busemann–Petty problem problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Busemann–Petty_problem
About this page
TagMV58A1F52E
URIm2://MV58A1F52E
Kindproblem
WritingW0 stub
MSC52A20, 52A40, 52A38
StatusThis copy of the site has no build data. Run m2 status MV58A1F52E.
AboutConvex body
UsesConvex body, Lebesgue measure

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.