Busemann–Petty problem

In the mathematical field of convex geometry, the Busemann–Petty problem, introduced by Herbert Busemann and Clinton Myers Petty (1956, problem 1), asks whether it is true that a symmetric convex body with larger central hyperplane sections has larger volume. More precisely, if K, T are symmetric convex bodies in Rⁿ such that

${\displaystyle \mathrm {Vol} _{n-1}\,(K\cap A)\leq \mathrm {Vol} _{n-1}\,(T\cap A)}$

for every hyperplane A passing through the origin, is it true that Vol_n K ≤ Vol_n T?

Busemann and Petty showed that the answer is positive if K is a ball. In general, the answer is positive in dimensions at most 4, and negative in dimensions at least 5.

History

Larman and

, and showed that the Busemann–Petty problem has a positive solution in a given dimension if and only if every symmetric convex body is an intersection body. An intersection body is a star body whose radial function in a given direction u is the volume of the hyperplane section u^⊥ ∩ K for some fixed star body K. ) then showed that the Busemann–Petty problem has a positive solution in dimension 4. Richard J. Gardner, A. Koldobsky, and T. Schlumprecht (1999) gave a uniform solution for all dimensions.

See also

• Shephard's problem

References

• MR 0950983
• Busemann, Herbert; Petty, Clinton Myers (1956), "Problems on convex bodies", Mathematica Scandinavica, 4: 88–94,
  MR 0084791, archived from the original
  on 2011-08-25
• Gardner, Richard J. (1994), "A positive answer to the Busemann-Petty problem in three dimensions",
  MR 1298719
• Gardner, Richard J.; Koldobsky, A.; Schlumprecht, Thomas B. (1999), "An analytic solution to the Busemann-Petty problem on sections of convex bodies",
  MR 1689343
• Koldobsky, Alexander (1998a), "Intersection bodies, positive definite distributions, and the Busemann-Petty problem",
  MR 1637955
• Koldobsky, Alexander (1998b), "Intersection bodies in R⁴",
  MR 1623669
• Koldobsky, Alexander (2005), Fourier analysis in convex geometry, Mathematical Surveys and Monographs, vol. 116, Providence, R.I.:
  MR 2132704
• Larman, D. G.; Rogers, C. A. (1975), "The existence of a centrally symmetric convex body with central sections that are unexpectedly small",
  MR 0390914
• MR 0963487
• Zhang, Gao Yong (1994), "Intersection bodies and the Busemann-Petty inequalities in R⁴",
  MR 1298716
  , The result in this paper is wrong; see the author's 1999 correction.
• Zhang, Gaoyong (1999), "A positive solution to the Busemann-Petty problem in R⁴",
  MR 1689339