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Displaying similar documents to “Sets invariant under projections onto two dimensional subspaces”

Sets invariant under projections onto one dimensional subspaces

Simon Fitzpatrick, Bruce Calvert (1991)

Commentationes Mathematicae Universitatis Carolinae

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The Hahn–Banach theorem implies that if m is a one dimensional subspace of a t.v.s. E , and B is a circled convex body in E , there is a continuous linear projection P onto m with P ( B ) ⊆ B . We determine the sets B which have the property of being invariant under projections onto lines through 0 subject to a weak boundedness type requirement.

On closed sets with convex projections in Hilbert space

Stoyu Barov, Jan J. Dijkstra (2007)

Fundamenta Mathematicae

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Let k be a fixed natural number. We show that if C is a closed and nonconvex set in Hilbert space such that the closures of the projections onto all k-hyperplanes (planes with codimension k) are convex and proper, then C must contain a closed copy of Hilbert space. In order to prove this result we introduce for convex closed sets B the set k ( B ) consisting of all points of B that are extremal with respect to projections onto k-hyperplanes. We prove that k ( B ) is precisely the intersection of...

A simple formula showing L¹ is a maximal overspace for two-dimensional real spaces

B. L. Chalmers, F. T. Metcalf (1992)

Annales Polonici Mathematici

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It follows easily from a result of Lindenstrauss that, for any real twodimensional subspace v of L¹, the relative projection constant λ(v;L¹) of v equals its (absolute) projection constant λ ( v ) = s u p X λ ( v ; X ) . The purpose of this paper is to recapture this result by exhibiting a simple formula for a subspace V contained in L ∞ ( ν ) and isometric to v and a projection P ∞ from C ⊕ V onto V such that ∥ P ∞ ∥ = ∥ P ₁ ∥ , where P₁ is a minimal projection from L¹(ν) onto v. Specifically, if P ₁ = ∑ i = 1 2 U i ⊗ v i , then P ∞ = ∑ i = 1 2 u i ⊗ V i , where d V i = 2 v i d ν and d U i = - 2 u i d ν .

A finite multiplicity Helson-Lowdenslager-de Branges theorem

Sneh Lata, Meghna Mittal, Dinesh Singh (2010)

Studia Mathematica

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We prove two theorems. The first theorem reduces to a scalar situation the well known vector-valued generalization of the Helson-Lowdenslager theorem that characterizes the invariant subspaces of the operator of multiplication by the coordinate function z on the vector-valued Lebesgue space L²(;ℂⁿ). Our approach allows us to prove an equivalent version of the vector-valued Helson-Lowdenslager theorem in a completely scalar setting, thereby eliminating the use of range functions and partial...

Circumradius versus side lengths of triangles in linear normed spaces

Gennadiy Averkov (2007)

Colloquium Mathematicae

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Given a planar convex body B centered at the origin, we denote by ℳ ²(B) the Minkowski plane (i.e., two-dimensional linear normed space) with the unit ball B. For a triangle T in ℳ ²(B) we denote by R B ( T ) the least possible radius of a Minkowskian ball enclosing T. We remark that in the terminology of location science R B ( T ) is the optimum of the minimax location problem with distance induced by B and vertices of T as existing facilities (see, for instance, [HM03] and the references therein)....