Affine group acting on hyperspaces of compact convex subsets of ℝⁿ

Sergey A. Antonyan; Natalia Jonard-Pérez

Displaying similar documents to “Affine group acting on hyperspaces of compact convex subsets of ℝⁿ”

Uniqueness of Cartesian Products of Compact Convex Sets

Zbigniew Lipecki, Viktor Losert, Jiří Spurný (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let $X_{i}$ , i∈ I, and $Y_{j}$ , j∈ J, be compact convex sets whose sets of extreme points are affinely independent and let φ be an affine homeomorphism of $\prod_{i \in I} X_{i}$ onto $\prod_{j \in J} Y_{j}$ . We show that there exists a bijection b: I → J such that φ is the product of affine homeomorphisms of $X_{i}$ onto $Y_{b (i)}$ , i∈ I.

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...

The natural operators of general affine connections into general affine connections

Jan Kurek, Włodzimierz M. Mikulski (2017)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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We reduce the problem of describing all $ℳ f_{m}$ -natural operators transforming general affine connections on $m$ -manifolds into general affine ones to the known description of all $G L (𝐑^{m})$ -invariant maps $𝐑^{m *} \otimes 𝐑^{m} \to \otimes^{k} 𝐑^{m *} \otimes \otimes^{k} 𝐑^{m}$ for $k = 1, 3$ .

On the Hausdorff dimension of certain self-affine sets

Abercrombie Alex G.., Nair R. (2002)

Studia Mathematica

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A subset E of ℝⁿ is called self-affine with respect to a collection ϕ₁,...,ϕₜ of affinities if E is the union of the sets ϕ₁(E),...,ϕₜ(E). For S ⊂ ℝⁿ let $Φ (S) = ⋃_{1 \leq j \leq t} ϕ_{j} (S)$ . If Φ(S) ⊂ S let $E_{Φ} (S)$ denote $⋂_{k \geq 0} Φ^{k} (S)$ . For given Φ consisting of contracting “pseudo-dilations” (affinities which preserve the directions of the coordinate axes) and subject to further mild technical restrictions we show that there exist self-affine sets $E_{Φ} (S)$ of each Hausdorff dimension between zero and a positive number depending on Φ. We also...

Pairs of convex bodies in a hyperspace over a Minkowski two-dimensional space joined by a unique metric segment

Agnieszka Bogdewicz, Jerzy Grzybowski (2009)

Banach Center Publications

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Let $(ℝ, | | \cdot | |_{)}$ be a Minkowski space with a unit ball and let $ϱ_{H}^{}$ be the Hausdorff metric induced by ${| | \cdot | |}_{}$ in the hyperspace of convex bodies (nonempty, compact, convex subsets of ℝ). R. Schneider [RSP] characterized pairs of elements of which can be joined by unique metric segments with respect to $ϱ_{H}^{B ⁿ}$ for the Euclidean unit ball Bⁿ. We extend Schneider’s theorem to the hyperspace $(², ϱ_{H}^{})$ over any two-dimensional Minkowski space.

On some results about convex functions of order $α$

M. Obradović, S. Owa (1986)

Matematički Vesnik

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The tame automorphism group of an affine quadric threefold acting on a square complex

Cinzia Bisi, Jean-Philippe Furter, Stéphane Lamy (2014)

Journal de l’École polytechnique — Mathématiques

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We study the group $Tame ({SL}_{2})$ of tame automorphisms of a smooth affine $3$ -dimensional quadric, which we can view as the underlying variety of ${SL}_{2} (ℂ)$ . We construct a square complex on which the group admits a natural cocompact action, and we prove that the complex is $CAT (0)$ and hyperbolic. We propose two applications of this construction: We show that any finite subgroup in $Tame ({SL}_{2})$ is linearizable, and that $Tame ({SL}_{2})$ satisfies the Tits alternative.

The Spaces of Closed Convex Sets in Euclidean Spaces with the Fell Topology

Katsuro Sakai, Zhongqiang Yang (2007)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let $C o n v_{F} (ℝ ⁿ)$ be the space of all non-empty closed convex sets in Euclidean space ℝ ⁿ endowed with the Fell topology. We prove that $C o n v_{F} (ℝ ⁿ) \approx ℝ ⁿ \times Q$ for every n > 1 whereas $C o n v_{F} (ℝ) \approx ℝ \times$ .

Some characterization of locally nonconical convex sets

Witold Seredyński (2004)

Czechoslovak Mathematical Journal

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A closed convex set $Q$ in a local convex topological Hausdorff spaces $X$ is called locally nonconical (LNC) if for every $x, y \in Q$ there exists an open neighbourhood $U$ of $x$ such that $(U \cap Q) + \frac{1}{2} (y - x) \subset Q$ . A set $Q$ is local cylindric (LC) if for $x, y \in Q$ , $x \neq y$ , $z \in (x, y)$ there exists an open neighbourhood $U$ of $z$ such that $U \cap Q$ (equivalently: $b d (Q) \cap U$ ) is a union of open segments parallel to $[x, y]$ . In this paper we prove that these two notions are equivalent. The properties LNC and LC were investigated in [3], where the implication $L N C \Rightarrow L C$ was proved in...

Minimal multi-convex projections

Grzegorz Lewicki, Michael Prophet (2007)

Studia Mathematica

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We say that a function from $X = C^{L} [0, 1]$ is k-convex (for k ≤ L) if its kth derivative is nonnegative. Let P denote a projection from X onto V = Πₙ ⊂ X, where Πₙ denotes the space of algebraic polynomials of degree less than or equal to n. If we want P to leave invariant the cone of k-convex functions (k ≤ n), we find that such a demand is impossible to fulfill for nearly every k. Indeed, only for k = n-1 and k = n does such a projection exist. So let us consider instead a more general “shape”...

On isometrical extension properties of function spaces

Hisao Kato (2015)

Commentationes Mathematicae Universitatis Carolinae

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In this note, we prove that any “bounded” isometries of separable metric spaces can be represented as restrictions of linear isometries of function spaces $C (Q)$ and $C (Δ)$ , where $Q$ and $Δ$ denote the Hilbert cube ${[0, 1]}^{\infty}$ and a Cantor set, respectively.

On isomorphisms between fibrations over $T^{k}$ associated with a $T^{k}$ action

Michał Sadowski (1991)

Annales Polonici Mathematici

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Separated sequences in uniformly convex Banach spaces

J. M. A. M. van Neerven (2005)

Colloquium Mathematicae

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We give a characterization of uniformly convex Banach spaces in terms of a uniform version of the Kadec-Klee property. As an application we prove that if (xₙ) is a bounded sequence in a uniformly convex Banach space X which is ε-separated for some 0 < ε ≤ 2, then for all norm one vectors x ∈ X there exists a subsequence $(x_{n_{j}})$ of (xₙ) such that $i n f_{j \neq k} | | x - (x_{n_{j}} - x_{n_{k}}) | | \geq 1 + δ_{X} (2 / 3 ε)$ , where $δ_{X}$ is the modulus of convexity of X. From this we deduce that the unit sphere of every infinite-dimensional uniformly convex Banach space...

Convex integration with constraints and applications to phase transitions and partial differential equations

Stefan Müller, Vladimír Šverák (1999)

Journal of the European Mathematical Society

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We study solutions of first order partial differential relations $D u \in K$ , where $u : Ω \subset ℝ^{n} \to ℝ^{m}$ is a Lipschitz map and $K$ is a bounded set in $m \times n$ matrices, and extend Gromov’s theory of convex integration in two ways. First, we allow for additional constraints on the minors of $D u$ and second we replace Gromov’s $P$ −convex hull by the (functional) rank-one convex hull. The latter can be much larger than the former and this has important consequences for the existence of ‘wild’ solutions to elliptic systems. Our...

Poincaré Inequalities and Moment Maps

Bo’az Klartag (2013)

Annales de la faculté des sciences de Toulouse Mathématiques

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We discuss a method for obtaining Poincaré-type inequalities on arbitrary convex bodies in $ℝ^{n}$ . Our technique involves a dual version of Bochner’s formula and a certain moment map, and it also applies to some non-convex sets. In particular, we generalize the central limit theorem for convex bodies to a class of non-convex domains, including the unit balls of $ℓ_{p}$ -spaces in $ℝ^{n}$ for $0 < p < 1$ .

Extending Maps in Hilbert Manifolds

Piotr Niemiec (2012)

Bulletin of the Polish Academy of Sciences. Mathematics

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Certain results on extending maps taking values in Hilbert manifolds by maps which are close to being embeddings are presented. Sufficient conditions on a map under which it is extendable by an embedding are given. In particular, it is shown that if X is a completely metrizable space of topological weight not greater than α ≥ ℵ₀, A is a closed set in X and f: X → M is a map into a manifold M modelled on a Hilbert space of dimension α such that $f (X ∖ A) \cap \bar{f (\partial A)} = \emptyset$ , then for every open cover of M there...

The equivariant universality and couniversality of the Cantor cube

Michael G. Megrelishvili, Tzvi Scarr (2001)

Fundamenta Mathematicae

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Let ⟨G,X,α⟩ be a G-space, where G is a non-Archimedean (having a local base at the identity consisting of open subgroups) and second countable topological group, and X is a zero-dimensional compact metrizable space. Let $⟨ H ({0, 1}^{ℵ ₀}), {0, 1}^{ℵ ₀}, τ ⟩$ be the natural (evaluation) action of the full group of autohomeomorphisms of the Cantor cube. Then (1) there exists a topological group embedding $φ : G ↪ H ({0, 1}^{ℵ ₀})$ ; (2) there exists an embedding $ψ : X ↪ {0, 1}^{ℵ ₀}$ , equivariant with respect to φ, such that ψ(X) is an equivariant retract of ${0, 1}^{ℵ ₀}$ with respect...

Countably convex $G_{δ}$ sets

Vladimir Fonf, Menachem Kojman (2001)

Fundamenta Mathematicae

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We investigate countably convex $G_{δ}$ subsets of Banach spaces. A subset of a linear space is countably convex if it can be represented as a countable union of convex sets. A known sufficient condition for countable convexity of an arbitrary subset of a separable normed space is that it does not contain a semi-clique [9]. A semi-clique in a set S is a subset P ⊆ S so that for every x ∈ P and open neighborhood u of x there exists a finite set X ⊆ P ∩ u such that conv(X) ⊈ S. For closed sets...