Finite-dimensional Normed Spaces Research Articles

Birkhoff–James classification of norm’s properties

For an arbitrary normed space X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {X}$$\\end{document} over a field F∈{R,C},\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {F}\\in \\{ \\mathbb {R}, \\mathbb {C}\\},$$\\end{document} we define the directed graph Γ(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma (\\mathcal {X})$$\\end{document} induced by Birkhoff–James orthogonality on the projective space P(X),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb P(\\mathcal {X}),$$\\end{document} and also its nonprojective counterpart Γ0(X).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma _0(\\mathcal {X}).$$\\end{document} We show that, in finite-dimensional normed spaces, Γ(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma (\\mathcal {X})$$\\end{document} carries all the information about the dimension, smooth points, and norm’s maximal faces. It also allows to determine whether the norm is a supremum norm or not, and thus classifies finite-dimensional abelian C∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^*$$\\end{document}-algebras among other normed spaces. We further establish the necessary and sufficient conditions under which the graph Γ0(R)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma _0({\\mathcal {R}})$$\\end{document} of a (real or complex) Radon plane R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {R}}$$\\end{document} is isomorphic to the graph Γ0(F2,‖·‖2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma _0(\\mathbb {F}^2, {\\Vert \\cdot \\Vert }_2)$$\\end{document} of the two-dimensional Hilbert space and construct examples of such nonsmooth Radon planes.

Some Results about Acts over Monoid and Bounded Linear Operators

This study delves into the properties of the associated act V over the monoid S of sinshT. It examines the relationship between faithful, finitely generated, and separated acts, as well as their connections to one-to-one and onto operators. Additionally, the correlation between acts over a monoid and modules over a ring is explored. Specifically, it is established that functions as an act over S if and only if functions as module, where T represents a nilpotent operator. Furthermore, it is proved that when T is onto operator and is finitely generated, is guaranteed to be finite-dimensional. Prove that for any bounded operator the following, is acting over S if and only if is a module where T is a nilpotent operator, is a faithful act over S, where T is any bounded linear operator, if T is any bounded operator, then is separated, if is separated act over S, Then T is injective, if a basis K = {vj, j} for V, then every element w of can be composed as =(pn (T) + . v, for some v in V, and put T as similar to any operator from to and V as a finite dimensional normed space, then is Noetherian act over S if S is Noetherian.

Metric Spaces where Geodesics are Never Unique

This article concerns a class of metric spaces, which we call multigeodesic spaces, where between any two distinct points there exist multiple distinct minimizing geodesics. We provide a simple characterization of multigeodesic normed spaces and deduce that is an example of such a space, but that finite-dimensional normed spaces are not. We also investigate what additional features are possible in arbitrary metric spaces which are multigeodesic.

On the existence of mass minimizing rectifiable G chains in finite dimensional normed spaces

We introduce the notion of density contractor of dimension m in a finite dimensional normed space X. If m+1=dim X this includes the area contracting projectors on hyperplanes whose existence was established by H. Busemann. If m=2, density contractors are an ersatz for such projectors and their existence, established here, follows from work by D. Burago and S. Ivanov. Once density contractors are available, the corresponding Plateau problem admits a solution among rectifiable G chains, regardless of the group of coefficients G. This is obtained as a consequence of the lower semicontinuity of the $m$ dimensional Hausdorff mass, of which we offer two proofs. One of these is based on a new type of integral geometric measure.

On the dimension of the set of minimal projections

Let X be a finite-dimensional normed space and let Y⊆X be its proper linear subspace. The set of all minimal projections from X to Y is a convex subset of the space all linear operators from X to X and we can consider its affine dimension. We establish several results on the possible values of this dimension. We prove optimal upper bounds in terms of the dimensions of X and Y. Moreover, we improve these estimates in the polyhedral normed spaces for an open and dense subset of subspaces of the given dimension. As a consequence, in the polyhedral normed spaces a minimal projection is unique for an open and dense subset of hyperplanes. To prove this, we establish certain new properties of the Chalmers-Metcalf operator. Another consequence is the fact, that for every subspace of a polyhedral normed space, there exists a minimal projection with many norming pairs.

Generalized orthogonality equations in finite-dimensional normed spaces

Let X, Y be real normed spaces and let rho '_+, rho '_- be norm derivatives. In this work, we solve a system of functional equations ρ+′(f(x),f(y))=g(x)ρ+′(x,y),ρ-′(f(x),f(y))=g(x)ρ-′(x,y),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll}\\rho '_+(f(x),f(y))=g(x)\\rho '_+(x,y),\\\\ \\rho '_-(f(x),f(y))=g(x)\\rho '_-(x,y), \\end{array}\\right. } \\end{aligned}$$\\end{document}with unknown functions f:X!rightarrow !Y, g:Xrightarrow mathbb {R}. Moreover, we give partial answer to open problem posed in 2010.

Horofunction Compactifications and Duality

We study the global topology and geometry of the horofunction compactification of classes of symmetric spaces under Finsler distances in three settings: bounded symmetric domains of the form B=B_1times cdots times B_r, where B_i is an open Euclidean ball in {mathbb {C}}^{n_i}, with the Kobayashi distance, symmetric cones with the Hilbert distance, and Euclidean Jordan algebras with the spectral norm. For these spaces we show, that the horofunction compactification is naturally homeomorphic to the closed unit ball of the dual norm of the Finsler metric in the tangent space at the basepoint. In each case we give an explicit homeomorphism. For finite dimensional normed spaces the link between the geometry of the horofunction compactification and the dual unit ball was suggested by Kapovich and Leeb, which we confirm for Euclidean Jordan algebras with the spectral norm. Our results also show that this duality phenomenon not only occurs in normed spaces, but also in a variety of noncompact type symmetric spaces with invariant Finsler metrics.

On some classical properties of normed spaces via generalized vector valued almost convergence

AbstractRecently, the authors interested some new problems on multiplier spaces of Lorentz’ almost convergence andfλ-convergence as a generalization of almost convergence.fλ-convergence is firstly introduced by Karakuş and Başar, and used for some new characterizations of completeness and barrelledness of the spaces through weakly unconditionally Cauchy series in a normed spaceXand its continuous dualX*. In the present paper, we deal withfλ-convergence to have some inclusion relations between the vector valued spaces obtained from this type convergence and corresponding classical sequence spaces, and to give new characterizations of some classical properties like completeness, reflexivity, Schur property and Grothendieck property of normed spaces. By the way, we give a characterization of finite-dimensional normed spaces.

Curvature measures and soap bubbles beyond convexity

Extending the celebrated results of Alexandrov (1958) and Korevaar–Ros (1988) for smooth sets, as well as the results of Schneider (1979) and the first author (1999) for arbitrary convex bodies, we obtain for the first time the characterization of the isoperimetric sets of a uniformly convex smooth finite-dimensional normed space (i.e. Wulff shapes) in the non-smooth and non-convex setting, based on a natural geometric condition involving the curvature measures. More specifically we show, under a weak mean convexity assumption, that finite unions of disjoint Wulff shapes are the only sets of positive reach A⊆Rn+1 with finite and positive volume such that, for some k∈{0,…,n−1}, the k-th generalized curvature measure Θkϕ(A,⋅), which is defined on the unit normal bundle of A with respect to the relative geometry induced by ϕ, is proportional to the top order generalized curvature measure Θnϕ(A,⋅). If k=n−1 the conclusion holds for all sets of positive reach with finite and positive volume. We also prove a related sharp result about the removability of the singularities. This result is based on the extension of the notion of a normal boundary point, originally introduced by Busemann and Feller (1936) for arbitrary convex bodies, to sets of positive reach.These findings are new even in the Euclidean space.Several auxiliary and related results are proved, which are of independent interest. They include the extension of the classical Steiner–Weyl tube formula to arbitrary closed sets in a finite dimensional uniformly convex normed vector space, a general formula for the derivative of the localized volume function, which extends and complements recent results of Chambolle–Lussardi–Villa (2021), and general versions of the Heintze–Karcher inequality.

Consensus in the Hegselmann–Krause Model

This paper is concerned with the probability of consensus in a multivariate, spatially explicit version of the Hegselmann-Krause model for the dynamics of opinions. Individuals are located on the vertices of a finite connected graph representing a social network, and are characterized by their opinion, with the set of opinions $\Delta$ being a general bounded convex subset of a finite dimensional normed vector space. Having a confidence threshold $\tau$, two individuals are said to be compatible if the distance (induced by the norm) between their opinions does not exceed the threshold $\tau$. Each vertex $x$ updates its opinion at rate the number of its compatible neighbors on the social network, which results in the opinion at $x$ to be replaced by a convex combination of the opinion at $x$ and the nearby opinions: $\alpha$ times the opinion at $x$ plus $(1 - \alpha)$ times the average opinion of its compatible neighbors. The main objective is to derive a lower bound for the probability of consensus when the opinions are initially independent and identically distributed with values in the opinion set $\Delta$.

About the continuity of one operation with convex compacts in finite-dimensional normed spaces

About the continuity of one operation with convex compacts in finite-dimensional normed spaces

An average John theorem

We prove that the $\frac12$-snowflake of a finite-dimensional normed space $(X,\|\cdot\|_X)$ embeds into a Hilbert space with quadratic average distortion $$O\Big(\sqrt{\log \mathrm{dim}(X)}\Big).$$ We deduce from this (optimal) statement that if an $n$-vertex expander embeds with average distortion $D\geqslant 1$ into $(X,\|\cdot\|_X)$, then necessarily $\mathrm{dim}(X)\geqslant n^{\Omega(1/D)}$, which is sharp by the work of Johnson, Lindenstrauss and Schechtman (1987). This improves over the previously best-known bound $\mathrm{dim}(X)\gtrsim (\log n)^2/D^2$ of Linial, London and Rabinovich (1995), strengthens a theorem of Matou\v{s}ek (1996) which resolved questions of Johnson and Lindenstrauss (1982), Bourgain (1985) and Arias-de-Reyna and Rodr{\'{\i}}guez-Piazza (1992), and answers negatively a question that was posed (for algorithmic purposes) by Andoni, Nguyen, Nikolov, Razenshteyn and Waingarten (2016).

When are maps preserving semi-inner products linear?

We observe that every map between finite-dimensional normed spaces of the same dimension that respects fixed semi-inner products must be automatically a linear isometry. Moreover, we construct a uniformly smooth renorming of the Hilbert space ell _2 and a continuous injection acting thereon that respects the semi-inner products, yet it is non-linear. This demonstrates that there is no immediate extension of the former result to infinite dimensions, even under an extra assumption of uniform smoothness.

Regular random sections of convex bodies and the random Quotient-of-Subspace Theorem

It was shown by G. Pisier that any finite-dimensional normed space admits an α-regular M-position, guaranteeing not only regular entropy estimates but moreover regular estimates on the diameters of minimal sections of its unit-ball and its dual. We revisit Pisier's argument and show the existence of a different position, which guarantees the same estimates for randomly sampled sections with high-probability. As an application, we obtain a random version of V. Milman's Quotient-of-Subspace Theorem, asserting that in the above position, typical quotients of subspaces are isomorphic to Euclidean, with a distance estimate which matches the best-known deterministic one (and beating all prior estimates which hold with high-probability). Our main novel ingredient is a new position of convex bodies, whose existence we establish by using topological arguments and a fixed-point theorem.

Max-plus convexity in Archimedean Riesz spaces

We study the topological properties of max-plus convex sets in an Archimedean Riesz space $E$ with respect to the topology and the max-plus structure associated to a given order unit $\boldsymbol u$; the definition of max-plus convex sets is algebraic and we do not assume that $E$ has an {\it a priori} given topological structure. To a given unit $\un$ one can associate two equivalent norms on $E$ one of which, denoted $\|\cdot\|_{\boldsymbol u}$, is classical, the other $\|\cdot\|_{\hun}$ is introduced here following a previous unpublished work of Stéphane Gaubert on the geodesic structure of finite dimensional max-plus; it is shown that the distance ${\sf D}_{\boldsymbol h\boldsymbol u}$ on $E$ associated to $\|\cdot\|_{\boldsymbol h\boldsymbol u}$ is a geodesic distance, called the Hilbert affine distance associated to $\boldsymbol u$, for which max-plus convex sets in $E$ are precisely the geodesically closed sets. Under suitable assumptions, we establish max-plus versions of some fixed points and continuous selection theorems that are well known for linear convex sets and we show that hyperspaces of compact max-plus convex sets are Absolute Retracts. We formulate a max-plus version of the Knaster-Kuratowski-Mazurkiewicz Lemma from which, following A. Granas and J. Dugundji, all of the consequences of the classical KKM Lemma can be derived in a max-plus version. P. de la Harpe showed that the interior of the standard simplex $\Delta_n$ equipped with the classical Hilbert metric-defined by the cross-ration of four appropriate points is isometric to a finite dimensional normed space. We give an explicit proof of that result: the norm space in question is $\mathbb R^n$ with the Hilbert affine norm $\|\cdot\|_{\boldsymbol h\boldsymbol u}$ with respect to $\boldsymbol u = (1, \ldots, 1)$.

Optimization of robot gripping forces and worst case loading

We consider the optimization of the vector of gripping forces that support a known generalized force acting on the gripped object—a rigid body or a mechanism. Working in the framework of finite dimensional normed vector spaces and their dual spaces, the cost function to be minimized is assumed to be a norm on the space of gripping forces. We present an expression for the optimum which depends on the external force and the kinematics of the gripping system. Next, assuming that optimal gripping forces are applied using force control, and assuming that there is a bound on the norm of the admissible gripping forces, we characterize the largest norm of an external force that the gripping system may support, that is, the norm of the worst case loading that may be applied and still be supported. A few simple examples are given for the sake of illustration.

Growth rate of Lipschitz constants for retractions between finite subset spaces

For any metric space $X$, finite subset spaces of $X$ provide a sequence of isometric embeddings $X=X(1)\subset X(2)\subset\cdots$. The existence of Lipschitz retractions $r_n\colon X(n)\to X(n-1)$ depends on the geometry of $X$ in a subtle way. Such retractions are known to exist when $X$ is an Hadamard space or a finite-dimensional normed space. But even in these cases it was unknown whether the sequence $\{r_n\}$ can be uniformly Lipschitz. We give a negative answer by proving that $\operatorname{Lip}(r_n)$ must grow with $n$ when $X$ is a normed space or an Hadamard space.

On sound ranging in proper metric spaces

We consider the sound ranging, or source localization, problem -- find the source-point from the moments when the wave-sphere of linearly, with time, increasing radius reaches the sensor-points -- in proper metric spaces (any closed ball is compact) and, in particular, in the finite-dimensional normed spaces. We approximate the solution to arbitrary precision by the iterative process with the stopping criterion. Implementation of the proposed method in Julia language is included.

Rigidity of symmetric frameworks in normed spaces

We develop a combinatorial rigidity theory for symmetric bar-joint frameworks in a general finite dimensional normed space. In the case of rotational symmetry, matroidal Maxwell-type sparsity counts are identified for a large class of d-dimensional normed spaces (including all ℓp spaces with p≠2). Complete combinatorial characterisations are obtained for half-turn rotation in the ℓ1 and ℓ∞-plane. As a key tool, a new Henneberg-type inductive construction is developed for the matroidal class of (2,2,0)-gain-tight gain graphs.

On a question of Pietch

The main result is that a finite dimensional normed space embeds isometrically in $\ell_p$ if and only if it has a discrete Levy $p$-representation. This provides an alternative answer to a question raised by Pietch, and as a corollary, a simple proof of the fact that unless $p$ is an even integer, the two-dimensional Hilbert space $\ell_2^2$ is not isometric to a subspace of $\ell_p$. The situation for $\ell_q^2$ with $q\neq 2$ turns out to be much more restrictive. The main result combined with a result of Dor provides a proof of the fact that if $q\neq 2$ then $\ell_q^2$ is not isometric to a subspace of $\ell_p$ unless $q=p$. Further applications concerning restrictions on the degree of smoothness of finite dimensional subspaces of $\ell_p$ are included as well.