arXiv subjects

Marcus Lõo

Publications and source records attributed to Marcus Lõo.

3 recordsLinked to original sources

arXiv2026

A Banach space with an unconditional basis which is not slicely countably determined

In this note, we study the geometry of the unit ball of the Banach space generated by the adequate family of all subsets of branches of the infinite binary tree, and answer several open questions related to slicely countably determined Banach spaces. Our main result is that the binary tree space is an example of a Banach space with an unconditional basis which fails to be slicely countably determined. In particular, it provides an example of a non slicely countably determined separable Banach space which contains no isomorphic copy of a space with the Daugavet property. We also exhibit some other geometric features of this space: we prove that its unit ball is dentable, that it has numerical index~1, and that the points of continuity of its unit ball form a weakly dense set. Finally, we show that the binary tree space contains a non-convex subset which is slicely countably determined, but does not admit a countable $π$-base for its relative weak topology, and that there is a 2-equivalent renorming of this space whose unit ball fails to be slicely countably determined.

math.FA↗

arXiv2025

The super Alternative Daugavet property for Banach spaces

We introduce the super alternative Daugavet property (super ADP) which lies strictly between the Daugavet property and the Alternative Daugavet property as follows. A Banach space $X$ has the super ADP if for every element $x$ in the unit sphere and for every relatively weakly open subset $W$ of the unit ball intersecting the unit sphere, one can find an element $y\in W$ and a modulus one scalar $θ$ such that $\|x+θy\|$ is almost two. It is known that spaces with the Daugavet property satisfy this condition, and that this condition implies the Alternative Daugavet property. We first provide examples of super ADP spaces which fail the Daugavet property. We show that the norm of a super ADP space is rough, hence the space cannot be Asplund, and we also prove that the space fails the point of continuity property (particularly, the Radon--Nikodým property). In particular, we get examples of spaces with the Alternative Daugavet property that fail the super ADP. For a better understanding of the differences between the super ADP, the Daugavet property, and the Alternative Daugavet property, we will also consider the localizations of these three properties and prove that they behave rather differently. As a consequence, we provide characterizations of the super ADP for spaces of vector-valued continuous functions and of vector-valued integrable functions.

math.FA↗

arXiv2024

Slicely countably determined points in Banach spaces

We introduce slicely countably determined points (SCD points) of a bounded and convex subset of a Banach space which extends the notions of denting points, strongly regular points and much more. We completely characterize SCD points in the unit balls of $L_1$-preduals. We study SCD points in direct sums of Banach spaces and obtain that an infinite sum of Banach spaces may have an SCD point despite the fact that none of its components have it. We then prove sufficient conditions to get that an elementary tensor $x\otimes y$ is an SCD point in the unit ball of the projective tensor product $X \widehat{\otimes}_πY$. Regarding Lipschitz-free spaces on compact metric spaces, we show that norm one SCD points of their unit balls are exactly the ones that can be approximated by convex combinations of strongly exposed points of the unit ball. Finally, as applications, we prove a new inheritance result for the Daugavet property to its subspaces, we show that separable Banach spaces for which every convex series of slices intersects the unit sphere must contain an isomorphic copy of $\ell_1$, and we get pointwise conditions on an operator on a Banach space with the Daugavet property to satisfy the Daugavet equation.

math.FA↗