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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.

A Chebyshev set is a subset of a normed linear space that admits unique best approximations. In the first part of this paper we present some basic results concerning Chebyshev sets. In particular, we investigate properties of the metric projection map, sufficient conditions for a subset of a normed linear space to be a Chebyshev set, and sufficient conditions for a Chebyshev set to be convex. In the second half of the paper we present a construction of a nonconvex Chebyshev subset of an inner product space.

We discuss Chebyshev sets of Riemannian manifolds. These are closed sets characterized by the existence of a well-defined distance-realizing projection onto them. The results we establish relate analytical properties of the distance function to these sets to their geometrical properties. They are extensions of some theorems on Chebyshev sets in Euclidean space to the context of Riemannian manifolds.

In this paper, we study convex analysis and its theoretical applications. We first apply important tools of convex analysis to Optimization and to Analysis. We then show various deep applications of convex analysis and especially infimal convolution in Monotone Operator  Theory. Among other things, we recapture the Minty surjectivity theorem in Hilbert space, and present a new proof of the sum theorem in reflexive spaces. More technically, we also discuss autoconjugate representers for maximally monotone operators. Finally, we consider various other applications in mathematical analysis.

We construct and analyze generalized Gaussian quadrature rules for integrands with endpoint singularities or near endpoint singularities. The rules have quadrature points inside the interval of integration, and the weights are all strictly positive. Such rules date back to the study of Chebyshev sets, but their use in applications has only recently been appreciated. We provide error estimates, and we show that the convergence rate is unaffected by the singularity of the integrand. We characterize the quadrature rules in terms of two families of functions that share many properties with orthogonal polynomials but that are orthogonal with respect to a discrete scalar product that, in most cases, is not known a priori.

Some new continuity concepts called Outer Radially Lower(ORL), Outer Radially Upper (ORU) and Inner Radially Lower (IRL), for set-valued metric projections were introduced in Banach spaces by B. Brosowski and F. Deutsch [Bull. Amer. Math. Soc. 78(1972), 974-978] to characterize suns and Chebyshev sets. In this paper we extend these concepts together with the concept of Inner Radially Upper (IRU) continuity to convex metric spaces and prove some results including those of Brosowski and Deutsch in such spaces.

A result on the existence and uniqueness of metric projection for certain sets is proved, by means of a saddle point theorem. A conjecture, based on such a result and aiming for the construction of a nonconvex Chebyshev set in a Hilbert space, is presented.

In this paper, we prove the following general result. Let XX be a real Hilbert space and J:X→RJ:X\\to \\textbf {R} a continuously Gâteaux differentiable, nonconstant functional, with compact derivative, such that \\[ lim sup‖x‖→+∞J(x)‖x‖2≤0 .\\limsup _{\\|x\\|\\to +\\infty }{{J(x)}\\over {\\|x\\|^2}}\\leq 0\\ . \\] Then, for each r∈ ]infXJ,supXJ[r\\in \\ ]\\inf _{X}J,\\sup _{X}J[ for which the set J−1([r,+∞[)J^{-1}([r,+\\infty [) is not convex and for each convex set S⊆XS\\subseteq X dense in XX, there exist x0∈S∩J−1(]−∞,r[)x_0\\in S\\cap J^{-1}(]-\\infty ,r[) and λ>0\\lambda >0 such that the equation \\[ x=λJ′(x)+x0x=\\lambda J’(x)+x_0 \\] has at least three solutions.

We derive the proximal normal formula for almost proximinal sets in a smooth and locally uniformly convex Banach space. Our technique leads us to show the generic Fréchet smoothness of the distance function in the case the norm is Fréchet smooth, and we derive a necessary and sufficient condition for the convexity of a Chebyshev set in a Banach space XX with norms on XX and X∗X^* locally uniformly convex.

Let Γ(X)\\Gamma (X) denote the proper, lower semicontinuous, convex functions on a Banach space XX, equipped with the completely metrizable topology τ\\tau of uniform convergence of distance functions on bounded sets. A function ff in Γ(X)\\Gamma (X) is called well-posed provided it has a unique minimizer, and each minimizing sequence converges to this minimizer. We show that well-posedness of f∈Γ(X)f \\in \\Gamma (X) is the minimal condition that guarantees strong convergence of approximate minima of τ\\tau-approximating functions to the minimum of ff. Moreover, we show that most functions in ⟨Γ(X),τaw⟩\\langle \\Gamma (X),{\\tau _{aw}}\\rangle are well-posed, and that this fails if Γ(X)\\Gamma (X) is topologized by the weaker topology of Mosco convergence, whenever XX is infinite dimensional. Applications to metric projections are also given, including a fundamental characterization of approximative compactness.