arXiv subjects
Publications and source records attributed to Jaan Kristjan Kaasik.
We construct equivalent norms on $L_1[0,1]$, arbitrarily close to the usual $L_1$-norm, which are weakly midpoint locally uniformly rotund and Gâteaux smooth, and whose dual norms have the Daugavet property. In particular, we obtain a strictly convex and smooth Banach space with the Daugavet property, thereby solving a longstanding open problem in the area.
We study universality of Lipschitz quotients. First, we modify a construction of Johnson, Lindenstrauss, Preiss and Schechtman to obtain a complete separable metric space that has every complete separable metric space as a Lipschitz quotient. Our main result is in the compact setting, where we prove that no such universal metric space can exist. We deduce this impossibility result by studying the curve-flat index, an ordinal index which provides a measure of the complexity of the curve-fragment structure in a metric space. We show that Lipschitz quotients cannot increase this index in compact domains; while there exist compact spaces with arbitrarily high countable curve-flat index. The main technical part of the paper is dedicated to proving a strong version of the latter fact: for every ordinal $α$ and every compact metric space $M$, there exists a compact metric space $N$ such that the curve-flat quotient of $N$ of order $α$ is almost-isometric to $M$.
We prove that in Lipschitz-free spaces the strong diameter two property, the diameter two property, and the local diameter two property coincide with their corresponding attaining variants.
We separate the local diameter two property from the diameter two property and their weak-star counterparts from each other in spaces of Lipschitz functions. We characterise the $w^*$-LD$2$P, the $w^*$-D$2$P, the LD$2$P, and the SD$2$P in these spaces. We introduce a generalised version of cyclical monotonicity to study functionals on spaces of Lipschitz functions.
We address some open problems concerning Banach spaces of real-valued Lipschitz functions. Specifically, we prove that the diameter two properties differ from their weak-star counterparts in these spaces. In particular, we establish the existence of dual Banach spaces lacking the symmetric strong diameter two property but possessing its weak-star counterpart. We show that there exists an octahedral Lipschitz-free space whose bidual is not octahedral. Furthermore, we prove that the Banach space of real-valued Lipschitz functions from any infinite subset of $\ell_1$ possesses the symmetric strong diameter two property. These results are achieved by introducing new sufficient conditions, providing new examples and clarifying the status of known ones.
We construct a Lipschitz-free space that is locally almost square but not weakly almost square; this is the first example of such a Banach space. We also prove a result, which indicates that geodesic metric spaces are a potential metric characterization for weakly almost square Lipschitz-free spaces. Lastly, we prove that a Lipschitz-free space can not have the symmetric strong diameter $2$ property.
We prove that the Lipschitz-free space over a metric space M is locally almost square whenever M is a length space. Consequently, the Lipschitz-free space is locally almost square if and only if it has the Daugavet property. We also show that a Lipschitz-free space is never almost square.