scholarly journals Relatively weakly open convex combinations of slices

2018 ◽  
Vol 146 (10) ◽  
pp. 4421-4427 ◽  
Author(s):  
Trond A. Abrahamsen ◽  
Vegard Lima
Keyword(s):  
Open Convex

scholarly journals An elementary proof of Minkowski's second inequality

1969 ◽  
Vol 10 (1-2) ◽  
pp. 177-181 ◽  
Author(s):  
I. Danicic
Keyword(s):  
Euclidean Space ◽  
Elementary Proof ◽  
Content Volume ◽  
Lattice Points ◽  
The Body ◽  
Dimensional Euclidean Space ◽  
Open Convex ◽  
Successive Minima ◽  
Linearly Independent ◽  
Independent Lattice

Let K be an open convex domain in n-dimensional Euclidean space, symmetric about the origin O, and of finite Jordan content (volume) V. With K are associated n positive constants λ1, λ2,…,λn, the ‘successive minima of K’ and n linearly independent lattice points (points with integer coordinates) P1, P2, …, Pn (not necessarily unique) such that all lattice points in the body λ,K are linearly dependent on P1, P2, …, Pj-1. The points P1,…, Pj lie in λK provided that λ > λj. For j = 1 this means that λ1K contains no lattice point other than the origin. Obviously

Equidistant Loci and the Minkowskian Geometries

1972 ◽  
Vol 24 (2) ◽  
pp. 312-327 ◽  
Author(s):  
B. B. Phadke
Keyword(s):  
Metric Spaces ◽  
Open Convex ◽  
Minkowskian Space ◽  
Geometric Conditions ◽  
Finite Dimensional ◽  
Projective Metric ◽  
Affine Line ◽  
Finite Dimensional Banach Space ◽  
Homogeneous Norm ◽  
Symmetric Distance

The spaced of this paper is a metrization, with a not necessarily symmetric distance xy, of an open convex set D in the n-dimensional affine space An such that xy + yz = xz if and only if x, y, z lie on an affine line with y between x and z and such that all the balls px ≦ p are compact. These spaces are called straight desarguesian G-spaces or sometimes open projective metric spaces. The hyperbolic geometry is an example; a large variety of other examples is studied by contributors to Hilbert's problem IV. When D = An and all the affine translations are isometries for the metric xy, the space is called a Minkowskian space or sometimes a finite dimensional Banach space, the (not necessarily symmetric) distance of a Minkowskian space being a (positive homogeneous) norm. In this paper geometric conditions in terms of equidistant loci are given for the space R to be a Minkowskian space.

scholarly journals Holomorphic extension of generalizations ofHpfunctions

1985 ◽  
Vol 8 (3) ◽  
pp. 417-424 ◽  
Author(s):  
Richard D. Carmichael
Keyword(s):  
Convex Hull ◽  
Extension Theorem ◽  
Holomorphic Functions ◽  
Holomorphic Extension ◽  
Finite Union ◽  
Convex Cones ◽  
Recent Analysis ◽  
Boundary Values ◽  
Connected Component ◽  
Open Convex

In recent analysis we have defined and studied holomorphic functions in tubes inℂnwhich generalize the HardyHpfunctions in tubes. In this paper we consider functionsf(z),z=x+iy, which are holomorphic in the tubeTC=ℝn+iC, whereCis the finite union of open convex conesCj,j=1,…,m, and which satisfy the norm growth of our new functions. We prove a holomorphic extension theorem in whichf(z),z ϵ TC, is shown to be extendable to a function which is holomorphic inT0(C)=ℝn+i0(C), where0(C)is the convex hull ofC, if the distributional boundary values in𝒮′off(z)from each connected componentTCjofTCare equal.

scholarly journals Three proofs of Minkowski's second inequality in the geometry of numbers

1965 ◽  
Vol 5 (4) ◽  
pp. 453-462 ◽  
Author(s):  
R. P. Bambah ◽  
Alan Woods ◽  
Hans Zassenhaus
Keyword(s):  
Convex Set ◽  
Geometry Of Numbers ◽  
Real Numbers ◽  
Open Convex ◽  
Positive Real ◽  
Original Proof ◽  
Linearly Independent ◽  
Image Position

Let K be a bounded, open convex set in euclidean n-space Rn, symmetric in the origin 0. Further let L be a lattice in Rn containing 0 and put extended over all positive real numbers ui for which uiK contains i linearly independent points of L. Denote the Jordan content of K by V(K) and the determinant of L by d(L). Minkowski's second inequality in the geometry of numbers states that Minkowski's original proof has been simplified by Weyl [6] and Cassels [7] and a different proof hasbeen given by Davenport [1].

scholarly journals A dual differentiation space without an equivalent locally uniformly rotund norm

2004 ◽  
Vol 77 (3) ◽  
pp. 357-364 ◽  
Author(s):  
Petar S. Kenderov ◽  
Warren B. Moors
Keyword(s):  
Dense Subset ◽  
Continuum Hypothesis ◽  
Open Convex ◽  
Residual Subset ◽  
Continuous Convex Function ◽  
Dual Differentiation ◽  
Fréchet Differentiable ◽  
Rotund Norm ◽  
Open Convex Subset ◽  
The Continuum

AbstractA Banach space (X, ∥ · ∥) is said to be a dual differentiation space if every continuous convex function defined on a non-empty open convex subset A of X* that possesses weak* continuous subgradients at the points of a residual subset of A is Fréchet differentiable on a dense subset of A. In this paper we show that if we assume the continuum hypothesis then there exists a dual differentiation space that does not admit an equivalent locally uniformly rotund norm.

Amenability and invariant subspaces

1974 ◽  
Vol 18 (2) ◽  
pp. 200-204 ◽  
Author(s):  
Anthony To-Ming Lau
Keyword(s):  
Convex Subset ◽  
Invariant Subspaces ◽  
Complex Field ◽  
Open Convex ◽  
Amenable Semigroup ◽  
Linear Action ◽  
Banach Theorem ◽  
Amenable Semigroups ◽  
Open Convex Subset ◽  
Invariant Element

Let E be a topological vector space (over the real or complex field). A well-known geometric form of the Hahn-Banach theorem asserts that if U is an open convex subset of E and M is a subspace of E which does not meet U, then there exists a closed hyperplane H containing M and not meeting U. In this paper we prove, among other things, that if S is a left amenable semigroup (which is the case, for example, when S is abelian or when S is a solvable group, see [3, p.11]), then for any right linear action of S on E, if U is an invariant open convex subset of E containing an invariant element and M is an invariant subspace not meeting U, then there exists a closed invariant hyperplane H of E containing M and not meeting U. Furthermore, this geometric property characterizes the class of left amenable semigroups.

scholarly journals Optimal boundary regularity for some singular Monge-Ampère equations on bounded convex domains

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Nam Q. Le
Keyword(s):  
Boundary Data ◽  
Boundary Regularity ◽  
Hölder Regularity ◽  
Convex Domains ◽  
Open Convex ◽  
Optimal Boundary ◽  
Monge Ampere ◽  
Holder Regularity ◽  
Bounded Convex

<p style='text-indent:20px;'>By constructing explicit supersolutions, we obtain the optimal global Hölder regularity for several singular Monge-Ampère equations on general bounded open convex domains including those related to complete affine hyperbolic spheres, and proper affine hyperspheres. Our analysis reveals that certain singular-looking equations, such as <inline-formula><tex-math id="M1">\begin{document}$ \det D^2 u = |u|^{-n-2-k} (x\cdot Du -u)^{-k} $\end{document}</tex-math></inline-formula> with zero boundary data, have unexpected degenerate nature.</p>

scholarly journals Rigidity of complex convex divisible sets

2018 ◽  
Vol 10 (04) ◽  
pp. 817-851
Author(s):  
Andrew M. Zimmer
Keyword(s):  
Projective Space ◽  
Complex Projective Space ◽  
Discrete Group ◽  
Convex Sets ◽  
Convex Set ◽  
Real Projective Space ◽  
Open Convex ◽  
Strictly Convex ◽  
Complex Projective

An open convex set in real projective space is called divisible if there exists a discrete group of projective automorphisms which acts cocompactly. There are many examples of such sets and a theorem of Benoist implies that many of these examples are strictly convex, have [Formula: see text] boundary, and have word hyperbolic dividing group. In this paper we study a notion of convexity in complex projective space and show that the only divisible complex convex sets with [Formula: see text] boundary are the projective balls.

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