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K-UR, K-LUR and K-R are the generalizations of UR, LUR and R respectively, which are of great significance in Banach space theory. While in Orlicz–Lorentz function space Λ φ , ω ∘ [ 0 , γ ) equipped with the Orlicz norm, the research methods of K-UR, K-LUR and K-R are much more complicated than those of UR, LUR and R. In this paper we obtain some criteria of K-UR, K-LUR and K-R of Λ φ , ω ∘ [ 0 , γ ) by means of the norm of dual space and H μ property of Λ φ , ω ∘ [ 0 , γ ) .

A nonreflexive Banach space may have a weakly uniformly rotund dual. The aim of this paper is to determine alternative characterisations and study further implications of this property in higher duals.

In this study, we conduct a literature review on normed linear spaces whose strengths are between rotundity and uniform rotundity. In this discourse, we also explore inter-relationships and juxtapositions between the subjects under consideration. There has been some discussion on the extent to which the geometry of the factor spaces has an impact on the geometry of the product spaces, as well as the degree to which the quotient spaces and subspaces inherit the geometry of the space itself. A comprehensive review has been conducted on the applications of most of these rotundities to some fields within the realm of approximation theory. In addition, some open problems are enumerated in the paper.

This paper mainly intends to investigate some geometric properties of Taylor sequence spaces. We examine the uniform rotundity, uniform smoothness, and the Radon-Riesz property of these spaces. Our results provide new examples of sequence spaces with these properties, although they are not isomorphic to ℓ p spaces.

Working constructively throughout, we introduce the notion of sufficient convexity for functions and sets and study its implications on the existence of best approximations of points in sets and of sets mutually.

We characterize rotund, uniformly rotund, locally uniformly rotund and compactly locally uniformly rotund spaces in terms of sets of (almost) farthest points from the unit sphere using the generalized diameter. For this, we introduce few remotality properties using the sets of almost farthest points. As a consequence, we obtain some characterizations of the aforementioned rotundity properties in terms of existing proximinality notions.

In this paper, the expression of Orlicz norm in Orlicz–Lorentz space generated by arbitrary Orlicz functions is given. The criteria for extreme points, strongly extreme points, as well as the sufficient and necessary conditions for mid-point locally uniform rotundity in Orlicz–Lorentz function spaces equipped with Orlicz norm generated by arbitrary Orlicz functions are given.

In this paper, the criterion that points of Orlicz function spaces equipped with Φ-Amemiya norm generated by an Orlicz function are strongly extreme is given. As a corollary, the sufficient and necessary conditions of midpoint local uniform rotundity of Orlicz function spaces equipped with Φ-Amemiya norm are obtained.

Given two (real) normed (linear) spaces $X$ and $Y$ , let $X\\otimes _{1}Y=(X\\otimes Y,\\Vert \\cdot \\Vert )$ , where $\\Vert (x,y)\\Vert =\\Vert x\\Vert +\\Vert y\\Vert$ . It is known that $X\\otimes _{1}Y$ is $2$ -UR if and only if both $X$ and $Y$ are UR (where we use UR as an abbreviation for uniformly rotund). We prove that if $X$ is $m$ -dimensional and $Y$ is $k$ -UR, then $X\\otimes _{1}Y$ is $(m+k)$ -UR. In the other direction, we observe that if $X\\otimes _{1}Y$ is $k$ -UR, then both $X$ and $Y$ are $(k-1)$ -UR. Given a monotone norm $\\Vert \\cdot \\Vert _{E}$ on $\\mathbb{R}^{2}$ , we let $X\\otimes _{E}Y=(X\\otimes Y,\\Vert \\cdot \\Vert )$ where $\\Vert (x,y)\\Vert =\\Vert (\\Vert x\\Vert _{X},\\Vert y\\Vert _{Y})\\Vert _{E}$ . It is known that if $X$ is uniformly rotund in every direction, $Y$ has the weak fixed point property for nonexpansive maps (WFPP) and $\\Vert \\cdot \\Vert _{E}$ is strictly monotone, then $X\\otimes _{E}Y$ has WFPP. Using the notion of $k$ -uniform rotundity relative to every $k$ -dimensional subspace we show that this result holds with a weaker condition on $X$ .

We give the method of construction of normal but not strongly normal positive cones.