Abstract We consider a nilpotent Lie group with a bracket-generating distribution ℋ {\mathcal{ {\mathcal H} }} and an asymmetric left-invariant norm ∣ ⋅ ∣ K {| \cdot | }_{K} induced by a convex body K ⊆ R k K\subseteq {{\mathbb{R}}}^{k} containing 0 in its interior. In this study, we prove the existence of minimizers of the perimeter functional P K {P}_{K} associated with ∣ ⋅ ∣ K {| \cdot | }_{K} under a volume (Haar measure) constraint.

In approximation theory, the problems of finding an estimate of the best approximation through the structural properties of the approximated function are well known. The work is devoted to such problems in spaces with an asymmetric norm and sign-sensitive weights.

Abstract: We test a general theory of norm changes based on evidence that people will punish less hygienic others more strongly than more hygienic others. The theory concludes that such asymmetric punishment would result in hygiene norms becoming ever stricter. We argue that, because complaints about one’s behavior might lead to protest, norms might not always change because of such complaints. We conducted an online experiment ( N = 1,023 Swiss adults) using handwashing as the target behavior. We replicated the asymmetry in punishment intensities and found that the intensity of protests against complaints about one’s too unhygienic behavior approached the intensity of these complaints. We conclude that, while asymmetric punishment may drive norm change, protests may lead to norm stability.

By definition, a Chebyshev set is a set of existence and uniqueness, that is, any point has a unique best approximant from this set. We study properties of Chebyshev sets composed of finitely or infinitely many planes (closed affine subspaces, possibly degenerated to points). We show that a finite union of planes is a Chebyshev set if and only if is a Chebyshev plane. Under some conditions on a space or a set, we show that a countable union of planes is never a Chebyshev set (unless this union is a Chebyshev plane itself). As a corollary, we give the following partial answer to the famous Efimov-Stechkin-Klee problem on convexity of Chebyshev sets: in Hilbert spaces (and, more generally, in reflexive (CLUR)-spaces), an at most countable union of planes is a Chebyshev set if and only if this set is a Chebyshev plane. Results of this kind are obtained both in usual normed linear spaces and in spaces with asymmetric norm.

Spaces with asymmetric metric and asymmetric norm are considered. It is shown that any metrizable separable asymmetrically normed linear space [Formula: see text] can be isometrically isomorphic imbedded, as an affine linear manifold, into the classical space [Formula: see text] with uniform norm [Formula: see text]. A similar result is obtained for spaces of density [Formula: see text]. For spaces with asymmetric metric, it is shown that each such space of density [Formula: see text] is isometric to a part of the space [Formula: see text] with the asymmetric seminorm [Formula: see text], where [Formula: see text].

Let M be a differentiable manifold, TxM be its tangent space at x ∈ M and TM = {(x, y);x ∈ M;y ∈ TxM} be its tangent bundle. A C0-Finsler structure is a continuous function F : TM → [0, ∞) such that F(x, ⋅) : TxM → [0, ∞) is an asymmetric norm. In this work we introduce the Pontryagin type C0-Finsler structures, which are structures that satisfy the minimum requirements of Pontryagin’s maximum principle for the problem of minimizing paths. We define the extended geodesic field ℰ on the slit cotangent bundle T*M\0 of (M, F), which is a generalization of the geodesic spray of Finsler geometry. We study the case where ℰ is a locally Lipschitz vector field. We show some examples where the geodesics are more naturally represented by ℰ than by a similar structure on TM. Finally we show that the maximum of independent Finsler structures is a Pontryagin type C0-Finsler structure where ℰ is a locally Lipschitz vector field.

We prove that an asymmetric normed space is never a Baire space if the topology induced by the asymmetric norm is not equivalent to the topology of a norm. More precisely, we show that a biBanach asymmetric normed space is a Baire space if and only if it is isomorphic to its associated normed space.

Directional assessment of traffic flow extremes

Abstract We consider an asymmetric left-invariant norm ∥ ⋅ ∥ K {\|\cdot\|_{K}} in the first Heisenberg group ℍ 1 {\mathbb{H}^{1}} induced by a convex body K ⊂ ℝ 2 {K\subset\mathbb{R}^{2}} containing the origin in its interior. Associated to ∥ ⋅ ∥ K {\|\cdot\|_{K}} there is a perimeter functional, that coincides with the classical sub-Riemannian perimeter in case K is the closed unit disk centered at the origin of ℝ 2 {{\mathbb{R}}^{2}} . Under the assumption that K has C 2 {C^{2}} boundary with strictly positive geodesic curvature we compute the first variation formula of perimeter for sets with C 2 {C^{2}} boundary. The localization of the variational formula in the non-singular part of the boundary, composed of the points where the tangent plane is not horizontal, allows us to define a mean curvature function H K {H_{K}} out of the singular set. In the case of non-vanishing mean curvature, the condition that H K {H_{K}} be constant implies that the non-singular portion of the boundary is foliated by horizontal liftings of translations of ∂ ⁡ K {\partial K} dilated by a factor of 1 H K {\frac{1}{H_{K}}} . Based on this we can define a sphere 𝕊 K {\mathbb{S}_{K}} with constant mean curvature 1 by considering the union of all horizontal liftings of ∂ ⁡ K {\partial K} starting from ( 0 , 0 , 0 ) {(0,0,0)} until they meet again in a point of the vertical axis. We give some geometric properties of this sphere and, moreover, we prove that, up to non-homogeneous dilations and left-translations, they are the only solutions of the sub-Finsler isoperimetric problem in a restricted class of sets.

Following the theory of symmetrically connected T 0-quasi-metric spaces constructed lately, in the present investigation the authors introduce and study the localized version of symmetric connectedness under the name local symmetric connectedness, as a new approach to determining the degree of the asymmetry and symmetry of T 0-quasi-metric spaces. In this context, some properties and topological aspects of locally symmetrically connected T 0-quasi-metric spaces are discussed by presenting various observations and many concrete (counter)examples. Specifically, it is shown that symmetric connectedness coincides with the local symmetric connectedness if we deal with the T 0-quasi-metric induced by an asymmetric norm which associates the theory of quasi-metrics in asymmetric topology with functional analysis.

A $C^0$-Finsler structure is a continuous function $F:TM \rightarrow [0,\infty)$ defined on the tangent bundle of a differentiable manifold $M$ such that its restriction to each tangent space is an asymmetric norm. We use the convolution of $F$ with the standard mollifier in order to construct a mollifier smoothing of $F$, which is a one parameter family of Finsler structures $F_\varepsilon$ (of class $\mathit{C}^\infty$ on $TM\backslash 0$) that converges uniformly to $F$ on compact subsets of $TM$. We prove that when $F$ is a Finsler structure, then the Chern connection, the Cartan connection, the Hashiguchi connection, the Berwald connection and the flag curvature of $F_\varepsilon$ converges uniformly on compact subsets to the corresponding objects of $F$. As an application of this mollifier smoothing, we study examples of two-dimensional piecewise smooth Riemannian manifolds with nonzero total curvature on a line segment. We also indicate how to extend this study to the correspondent piecewise smooth Finsler manifolds.

Huber-type principal expectile component analysis

Pseudo-quantile functional data clustering

In the realms of theoretical computer science and approximation theory, asymetric normed spaces play an important role. In this paper, by combining asymmetric norm and cone norm, it is defined asymmetric cone normed spaces. Also, it is introduced and studied some topological concepts in asymmetric cone normed spaces.

The weak topology in finite dimensional asymmetric normed spaces

S. Hansen and E. Zuazua [SIAM J. Control Optim., 33 (1995), pp. 1357--1391] studied the problem of exact controllability of two strings connected by a point mass with constant physical coefficients. In this paper we study the same problem with variable physical coefficients. This system is generated by the following equations: $\rho(x) u_{tt}=(\sigma(x) u_{x})_{x}-q(x)u$, $x\in (-1,0)\cup (0,1)$, $t>0$, $Mu_{tt}(0,t)+\sigma_{1}(0)u_{x}(0^{-},t)-\sigma_{2}(0)u_{x}(0^{+},t)=0$, $t>0$, with a Dirichlet boundary condition on the left end and a control on the right end. We prove that this system is exactly controllable in an asymmetric space for the control time $T> 2\int_{-1}^{1}(\frac{\rho(x)}{\sigma(x)})^{\frac{1}{2}}dx$. We establish the equivalence between a suitable asymmetric norm of the initial data and the $L^{2}(0,T)$-norm of $u_{x}(1,t)$ (where $u$ is the solution of the uncontrolled system). Our approach is mainly based on a detailed spectral analysis and the theory of divided differences. In particular, we prove that the spectral gap $(\sqrt{\lambda_{n+1}}-\sqrt{\lambda_{n}})$ tends to zero of the order of $\frac{1}{n}$.

Principal component analysis in an asymmetric norm

Principal Component Analysis in an Asymmetric Norm

A special class of separated normed cones, which includes convex cones in normed spaces and in spaces with an asymmetric norm, is distinguished on the basis of the functional separability of elements. It is shown that, generally, separated normed cones admit no linear injective isometric embedding in any normed space. An analog of the Banach–Mazur theorem on a sublinear injective embedding of a separated normed cone in the cone of real nonnegative continuous functions on the interval [0; 1] with the ordinary sup-norm is obtained. This result is used to prove the existence of a countable total set of bounded linear functionals for a special class of separated normed cones.

An asymmetric norm is a positive sublinear functional p on a real vector space X satisfying \(x=\theta _X\) whenever \(p(x)=p(-x)=0\). Since the space of all lower semi-continuous linear functionals of an asymmetric normed space is not a linear space, the theory is different in the asymmetric case. The main purpose of this study is to define bounded and continuous linear operators acting between asymmetric cone normed spaces. After examining the differences with symmetric case, we give some results related to Baire’s characterization of completeness in asymmetric cone normed spaces.