Busemann–Petty Problem

In the mathematical field of convex geometry, the Busemann–Petty problem, introduced by , 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 R^''n'' such that

:$\mathrm_ \, (K \cap A) \leq \mathrm_ \, (T \cap A)$

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

showed that the Busemann–Petty problem has a negative solution in dimensions at least 12, and this bound was reduced to dimensions at least 5 by several other authors. pointed out a particularly simple counterexample: all sections of the unit volume cube have measure at most , while in dimensions at least 10 all central sections of the unit volume ball have measure at least . introduced intersection bodies, 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''.
used Lutwak's result to show that the Busemann–Petty problem has a positive solution if the dimension is 3. claimed incorrectly that the unit cube in R⁴ is not an intersection body, which would have implied that the Busemann–Petty problem has a negative solution if the dimension is at least 4. However showed that a centrally symmetric star-shaped body is an intersection body if and only if the function 1/, , ''x'', , is a positive definite distribution, where , , x, , is the homogeneous function of degree 1 that is 1 on the boundary of the body, and used this to show that the unit balls l, 1 < ''p'' ≤ ∞ in ''n''-dimensional space with the l^''p'' norm are intersection bodies for ''n'' = 4 but are not intersection bodies for ''n'' ≥ 5, showing that Zhang's result was incorrect. then showed that the Busemann–Petty problem has a positive solution in dimension 4.
gave a uniform solution for all dimensions.

See also

* Shephard's problem

References

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{{DEFAULTSORT:Busemann-Petty problem
Convex geometry