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Articles 1 - 5 of 5
Full-Text Articles in Physical Sciences and Mathematics
A Positive Answer To The Busemann-Petty Problem In 3 Dimensions, Richard J. Gardner
A Positive Answer To The Busemann-Petty Problem In 3 Dimensions, Richard J. Gardner
Mathematics Faculty Publications
We prove that in E3 the Busemann-Petty problem, concerning central sections of centrally symmetric convex bodies, has a positive answer. Together with other results, this settles the problem in each dimension.
Convex-Bodies With Similar Projections, Richard J. Gardner, Aljoša VolčIč
Convex-Bodies With Similar Projections, Richard J. Gardner, Aljoša VolčIč
Mathematics Faculty Publications
By examining an example constructed by Petty and McKinney, we show that there are pairs of centered and coaxial bodies of revolution in Ed, d ≥ 3, whose projections onto each two-dimensional subspace are similar, but which are not themselves even affinely equivalent.
On The Busemann-Petty Problem Concerning Central Sections Of Centrally Symmetric Convex-Bodies, Richard J. Gardner
On The Busemann-Petty Problem Concerning Central Sections Of Centrally Symmetric Convex-Bodies, Richard J. Gardner
Mathematics Faculty Publications
We present a method which shows that in E3 the Busemann-Petty problem, concerning central sections of centrally symmetric convex bodies, has a positive answer. Together with other results, this settles the problem in each dimension.
Intersection Bodies And The Busemann-Petty Problem, Richard J. Gardner
Intersection Bodies And The Busemann-Petty Problem, Richard J. Gardner
Mathematics Faculty Publications
It is proved that the answer to the Busemann-Petty problem concerning central sections of centrally symmetric convex bodies in d-dimensional Euclidean space Ed is negative for a given d if and only if certain centrally symmetric convex bodies exist in Ed which are not intersection bodies. It is also shown that a cylinder in Ed is an intersection body if and only if d ≤ 4, and that suitably smooth axis-convex bodies of revolution are intersection bodies when d ≤ 4. These results show that the Busemann-Petty problem has a negative answer for d ≥ 5 …
Operator Equations And Invariant Subspaces, Valentin Matache
Operator Equations And Invariant Subspaces, Valentin Matache
Mathematics Faculty Publications
Banach space operators acting on some fixed space X are considered. If two such operators A and B verify the condition A2 = B2 and if A has non-trivial hyperinvariant subspaces, then B has nontrivial invariant subspaces. If A and B commute and satisfy a special type of functional equation, and if A is not a scalar multiple of the identity, the author proves that if A has nontrivial invariant subspaces, then so does B.