Related papers: Regularity points and Jensen measures for $R(X)$

We present a construction that enables one to find Banach spaces $X$ whose sets $NA(X)$ of norm attaining functionals do not contain two-dimensional subspaces and such that, consequently, $X$ does not contain proximinal subspaces of finite…

A remarkable theorem of R. C. James is the following: suppose that $X$ is a Banach space and $C \subseteq X$ is a norm bounded, closed and convex set such that every linear functional $x^* \in X^*$ attains its supremum on $C$; then $C$ is a…

We construct a Banach space $X$ for which the set of norm-attaining functionals $NA(X,\mathbb{R})$ does not contain any non-trivial cone. Even more, given two linearly independent norm-attaining functionals on $X$, no other element of the…

This article generalizes the work of Ballmann and \'Swiatkowski to the case of Reflexive Banach spaces and uniformly convex Busemann spaces, thus giving a new fixed point criterion for groups acting on simplicial complexes.

In this article, we discuss a new version of metric fixed point theory especially of Banach Contraction Principle, Ran-Reurings Theorem and others.

In this article, we will characterize regular points respectively by the local vanishing, positivity of the Ricci curvature and $L^2$-solvability of the $\overline\partial$-equation together with Skoda's theorem for Nadel-Lebesgue…

We consider several notions of regularity, including strong regularity, bounded relative units, and Ditkin's condition, in the setting of vector-valued function algebras. Given a commutative Banach algebra $A$ and a compact space $X$, let…

We prove that any pair of reasonable cross norms defined on the tensor product of $n$ Banach spaces induce $(2k)^{n-1}$-Lipschitz equivalent metrics (and thus, a unique topology) on the set $S^k_{X_1,\ldots, X_n}$ of vectors of rank $\leq…

In this paper we consider the compact plane sets known as Swiss cheese sets, which are a useful source of examples in the theory of uniform algebras and rational approximation. We introduce a notion of 'allocation map' connected with Swiss…

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

We provide, in every infinite-dimensional separable Banach space, an average locally uniformly rotund (and hence rotund) Gateaux smooth renorming which is not locally uniformly rotund. This solves an open problem posed by A.J. Guirao, V.…

We provide regularity of solutions to a large class of evolution equations on Banach spaces where the generator is composed of a static principal part plus a non-autonomous perturbation. Regularity is examined with respect to the graph norm…

The paper elucidates the relationship between the density of a Banach space and possible sizes of well-separated subsets of its unit sphere. For example, it is proved that for a large enough space $X$, the unit sphere $S_X$ always contains…

In this paper we survey known results of characterizations of reflexive Banach spaces, which are based on convergence of usual and generalized arithmetic mean (or Ces\`aro sum), weakly compact subsets, affine sets in a Banach space or its…

We extend the theory of regularity structures [Hai14] to allow processes belonging to locally $m$-convex topological algebras. This extension includes processes in the locally $C^{*}$-algebras of [CHP25] used to localise singular stochastic…

Swiss cheese sets are compact subsets of the complex plane obtained by deleting a sequence of open disks from a closed disk. Such sets have provided numerous counterexamples in the theory of uniform algebras. In this paper, we introduce a…

A class of semi-bounded solutions of the two-dimensional incompressible Euler equations satisfying either periodic or Dirichlet boundary conditions is examined. For smooth initial data, new blowup criteria in terms of the initial concavity…

We introduce a new type of norm for ordered vector spaces majorized by a proper (convex) cone that generalizes the notions of order unit norm and base norm. Then we give sufficient conditions to ensure its completeness. In the case of…

We study existence and partial regularity relative to the weighted Steiner problem in Banach spaces. We show $C^1$ regularity almost everywhere for almost minimizing sets in uniformly rotund Banach spaces whose modulus of uniform convexity…

Various notions of stable ranks are studied for topological algebras. Some partial answers to R.G. Swan's problem (Have two Banach or good Fr\'echet algebras as in the density theorem in K-theory the same stable rank ?) are obtained. For…