On exposed points and extremal points of convex sets in ℝⁿ and Hilbert space

Stoyu Barov; Jan J. Dijkstra

Displaying similar documents to “On exposed points and extremal points of convex sets in ℝⁿ and Hilbert space”

Convexity of sublevel sets of plurisubharmonic extremal functions

Finnur Lárusson, Patrice Lassere, Ragnar Sigurdsson (1998)

Annales Polonici Mathematici

Similarity:

Let X be a convex domain in ℂⁿ and let E be a convex subset of X. The relative extremal function $u_{E, X}$ for E in X is the supremum of the class of plurisubharmonic functions v ≤ 0 on X with v ≤ -1 on E. We show that if E is either open or compact, then the sublevel sets of $u_{E, X}$ are convex. The proof uses the theory of envelopes of disc functionals and a new result on Blaschke products.

Siciak’s extremal function of convex sets in $C^{N}$

Mirosław Baran (1988)

Annales Polonici Mathematici

Similarity:

Extremal plurisubharmonic functions in $C^{N}$

Józef Siciak (1981)

Annales Polonici Mathematici

Similarity:

Extremal sections of complex $l_{p}$ -balls, 0 < p ≤ 2

Alexander Koldobsky, Marisa Zymonopoulou (2003)

Studia Mathematica

Similarity:

We study the extremal volume of central hyperplane sections of complex n-dimensional $l_{p}$ -balls with 0 < p ≤ 2. We show that the minimum corresponds to hyperplanes orthogonal to vectors ξ = (ξ¹,...,ξⁿ) ∈ ℂⁿ with |ξ¹| = ... = |ξⁿ|, and the maximum corresponds to hyperplanes orthogonal to vectors with only one non-zero coordinate.

On closed sets with convex projections in Hilbert space

Stoyu Barov, Jan J. Dijkstra (2007)

Fundamenta Mathematicae

Similarity:

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...

Properties of extremal sequences for the Bellman function of the dyadic maximal operator

Eleftherios N. Nikolidakis (2013)

Colloquium Mathematicae

Similarity:

We prove a necessary condition that has every extremal sequence for the Bellman function of the dyadic maximal operator. This implies the weak- $L^{p}$ uniqueness for such a sequence.

The logarithmic capacity in $C^{n}$

S. Kołodziej (1988)

Annales Polonici Mathematici

Similarity:

Universally divergent Fourier series via Landau's extremal functions

Gerd Herzog, Peer Chr. Kunstmann (2015)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

We prove the existence of functions $f \in A (𝔻)$ , the Fourier series of which being universally divergent on countable subsets of $𝕋 = \partial 𝔻$ . The proof is based on a uniform estimate of the Taylor polynomials of Landau’s extremal functions on $𝕋 ∖ {1}$ .

Extreme points of the complex binary trilinear ball

Fernando Cobos, Thomas Kühn, Jaak Peetre (2000)

Studia Mathematica

Similarity:

We characterize all the extreme points of the unit ball in the space of trilinear forms on the Hilbert space $ℂ^{2}$ . This answers a question posed by R. Grząślewicz and K. John [7], who solved the corresponding problem for the real Hilbert space $ℝ^{2}$ . As an application we determine the best constant in the inequality between the Hilbert-Schmidt norm and the norm of trilinear forms.

On some extremal problems in classes of holomorphic functions $S_{p}^{} (ρ), S_{p}^{\circ} (ρ), Σ_{p}^{} (ρ)$

B. Platynowicz (1980)

Matematički Vesnik

Similarity:

The postage stamp problem and arithmetic in base $r$

Amitabha Tripathi (2008)

Czechoslovak Mathematical Journal

Similarity:

Let $h, k$ be fixed positive integers, and let $A$ be any set of positive integers. Let $h A : = {a_{1} + a_{2} + \dots + a_{r} : a_{i} \in A, r \leq h}$ denote the set of all integers representable as a sum of no more than $h$ elements of $A$ , and let $n (h, A)$ denote the largest integer $n$ such that ${1, 2, ..., n} \subseteq h A$ . Let $n (h, k) : = {max}_{A} : n (h, A)$ , where the maximum is taken over all sets $A$ with $k$ elements. We determine $n (h, A)$ when the elements of $A$ are in geometric progression. In particular, this results in the evaluation of $n (h, 2)$ and yields surprisingly sharp lower bounds for $n (h, k)$ , particularly for $k = 3$ .

Siciak’s extremal function via Bernstein and Markov constants for compact sets in $ℂ^{N}$

Leokadia Bialas-Ciez (2012)

Annales Polonici Mathematici

Similarity:

The paper is concerned with the best constants in the Bernstein and Markov inequalities on a compact set $E \subset ℂ^{N}$ . We give some basic properties of these constants and we prove that two extremal-like functions defined in terms of the Bernstein constants are plurisubharmonic and very close to the Siciak extremal function $Φ_{E}$ . Moreover, we show that one of these extremal-like functions is equal to $Φ_{E}$ if E is a nonpluripolar set with $l i m_{n \to \infty} M ₙ {(E)}^{1 / n} = 1$ where $M ₙ (E) : = s u p {| | | g r a d P | | |}_{E} / {| | P | |}_{E}$ , the supremum is taken over all polynomials P of N variables...

Sharp estimation of the coefficients of bounded univalent functions close to identity

Lucjan Siewierski

Similarity:

CONTENTSIntroduction...............................................................................................................................................................................5Definitions and notation.........................................................................................................................................................7The main result........................................................................................................................................................................91....

On some results about convex functions of order $α$

M. Obradović, S. Owa (1986)

Matematički Vesnik

Similarity:

Addendum to “On Hilbert sets and $C_{λ} (g)$ -spaces with no subspace isomorphic to $c_{0}$ ”

Daniel Li (1995)

Colloquium Mathematicae

Similarity:

On the Kaczmarz algorithm of approximation in infinite-dimensional spaces

Stanisław Kwapień, Jan Mycielski (2001)

Studia Mathematica

Similarity:

The Kaczmarz algorithm of successive projections suggests the following concept. A sequence $(e_{k})$ of unit vectors in a Hilbert space is said to be effective if for each vector x in the space the sequence (xₙ) converges to x where (xₙ) is defined inductively: x₀ = 0 and $x ₙ = x_{n - 1} + α ₙ e ₙ$ , where $α ₙ = ⟨ x - x_{n - 1}, e ₙ ⟩$ . We prove the effectivity of some sequences in Hilbert spaces. We generalize the concept of effectivity to sequences of vectors in Banach spaces and we prove some results for this more general concept.

A classification of projectors

Gustavo Corach, Alejandra Maestripieri, Demetrio Stojanoff (2005)

Banach Center Publications

Similarity:

A positive operator A and a closed subspace of a Hilbert space ℋ are called compatible if there exists a projector Q onto such that AQ = Q*A. Compatibility is shown to depend on the existence of certain decompositions of ℋ and the ranges of A and $A^{1 / 2}$ . It also depends on a certain angle between A() and the orthogonal of .

On the separation properties of $K_{ω}$

Malgorzata Wójcicka (1986)

Colloquium Mathematicae

Similarity: