Busemann–Petty problem
Encyclopedia
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 Rn such that
for every hyperplane A passing through the origin, is it true that Voln K ≤ Voln 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.
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 lp 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 (giving an unusual example of two papers by the same author in the same leading journal claiming opposite solutions to a problem).
gave a uniform solution for all dimensions.
Convex geometry
Convex geometry is the branch of geometry studying convex sets, mainly in Euclidean space.Convex sets occur naturally in many areas of mathematics: computational geometry, convex analysis, discrete geometry, functional analysis, geometry of numbers, integral geometry, linear programming,...
, the Busemann–Petty problem, introduced by , asks whether it is true that a symmetric convex body
Convex body
In mathematics, a convex body in n-dimensional Euclidean space Rn is a compact convex set with non-empty interior.A convex body K is called symmetric if it is centrally symmetric with respect to the origin, i.e. a point x lies in K if and only if its antipode, −x, also lies in K...
with larger central hyperplane sections has larger volume. More precisely, if K, T are symmetric convex bodies in Rn such that
for every hyperplane A passing through the origin, is it true that Voln K ≤ Voln 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 √2, while in dimensions at least 10 all central sections of the unit volume ball have measure at least √2. 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 lp 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 (giving an unusual example of two papers by the same author in the same leading journal claiming opposite solutions to a problem).
gave a uniform solution for all dimensions.