arXiv subjects

Rihhard Nadel

Publications and source records attributed to Rihhard Nadel.

3 recordsLinked to original sources

arXiv2020

On Daugavet indices of thickness

Inspired by R. Whitley's thickness index the last named author recently introduced the Daugavet index of thickness of Banach spaces. We continue the investigation of the behavior of this index and also consider two new versions of the Daugavet index of thickness, which helps us solve an open problem which connect the Daugavet indices with the Daugavet equation. Moreover, we will improve the formerly known estimates of the behavior of Daugavet index on direct sums of Banach spaces by establishing sharp bounds. As a consequence of our results we prove that, for every $0<δ<2$, there exists a Banach space where the infimum of the diameter of convex combinations of slices of the unit ball is exactly $δ$, solving an open question from the literature. Finally, we prove that an open question posed by Ivakhno in 2006 about the relation between the radius and diameter of slices has a negative answer.

math.FA↗

arXiv2018

Symmetric strong diameter two property

We study Banach spaces with the property that, given a finite number of slices of the unit ball, there exists a direction such that all these slices contain a line segment of length almost 2 in this direction. This property was recently named the symmetric strong diameter two property by Abrahamsen, Nygaard, and Põldvere. The symmetric strong diameter two property is not just formally stronger than the strong diameter two property (finite convex combinations of slices have diameter 2). We show that the symmetric strong diameter two property is only preserved by $\ell_\infty$-sums, and working with weak star slices we show that $\text{Lip}_0(M)$ have the weak star version of the property for several classes of metric spaces $M$.

math.FA↗

arXiv2017

Stability of average roughness, octahedrality, and strong diameter 2 properties of Banach spaces with respect to absolute sums

We prove that, if Banach spaces $X$ and $Y$ are $δ$-average rough, then their direct sum with respect to an absolute norm $N$ is $δ/N(1,1)$-average rough. In particular, for octahedral $X$ and $Y$ and for $p$ in $(1,\infty)$ the space $X\oplus_p Y$ is $2^{1-1/p}$-average rough, which is in general optimal. Another consequence is that for any $δ$ in $(1,2]$ there is a Banach space which is exactly $δ$-average rough. We give a complete characterization when an absolute sum of two Banach spaces is octahedral or has the strong diameter 2 property. However, among all of the absolute sums, the diametral strong diameter 2 property is stable only for 1- and $\infty$-sums.

math.FA↗