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dc.contributor.author
Basso, Giuliano
dc.contributor.supervisor
Lang, Urs
dc.contributor.supervisor
Lytchak, Alexander
dc.contributor.supervisor
Naor, Assaf
dc.date.accessioned
2020-02-17T07:56:46Z
dc.date.available
2020-02-12T16:28:44Z
dc.date.available
2020-02-13T10:26:37Z
dc.date.available
2020-02-16T10:19:00Z
dc.date.available
2020-02-17T07:56:46Z
dc.date.issued
2019-12
dc.identifier.uri
http://hdl.handle.net/20.500.11850/398970
dc.identifier.doi
10.3929/ethz-b-000398970
dc.description.abstract
The subject of this doctoral thesis is the class of barycentric metric spaces, which encompasses both Banach spaces and complete CAT(0) spaces. Encouraged by known results as well as open questions in the context of CAT(0) spaces, we study similar objectives in the framework of barycentric metric spaces. For example, we show that certain fixed point properties, which are given in CAT(0) spaces, do not hold for some barycentric metric spaces, and prove two fixed point results adapted to the new situation. These results are phrased for the class of metric spaces that allow a conical bicombing; this is no restriction, since the class of barycentric metric spaces agrees with this class. This equality leads to a variety of questions regarding the existence and uniqueness of certain classes of conical bicombings. In particular, we consider conical bicombings on open subsets of normed vector spaces and show that these bicombings are locally given by linear segments. This result implies that any open convex subset in a large class of Banach spaces possesses a unique consistent conical bicombing. Besides this, we consider various Lipschitz extension problems, where in some cases any complete barycentric metric space may appear as target space. One such Lipschitz extension problem involves the extension of a Lipschitz function to finitely many additional points. Our contribution consists of finding upper bounds for the distortion of the Lipschitz constant, and we construct examples which demonstrate that we found the best possible bounds in the case of an extension to one additional point. Many Lipschitz extension constants may be computed by solving an associated linear extension problem, which is why, in the last part, we turn our attention to absolute linear projection constants of real Banach spaces. We succeeded in finding a formula for the maximal linear projection constant amongst \(n\)-dimensional Banach spaces. By means of this formula, we give another proof of the Grünbaum conjecture, which was first proven by Chalmers and Lewicki in 2010.
en_US
dc.format
application/pdf
en_US
dc.language.iso
en
en_US
dc.publisher
ETH Zurich
en_US
dc.rights.uri
http://rightsstatements.org/page/InC-NC/1.0/
dc.subject
Metric spaces
en_US
dc.subject
Projection constants
en_US
dc.subject
Lipschitz maps
en_US
dc.subject
two-graphs
en_US
dc.subject
Non-positive curvature
en_US
dc.title
Fixed point and Lipschitz extension theorems for barycentric metric spaces
en_US
dc.type
Doctoral Thesis
dc.rights.license
In Copyright - Non-Commercial Use Permitted
dc.date.published
2020-02-13
ethz.size
101 p.
en_US
ethz.code.ddc
DDC - DDC::5 - Science::510 - Mathematics
en_US
ethz.identifier.diss
26486
en_US
ethz.publication.place
Zurich
en_US
ethz.publication.status
published
en_US
ethz.leitzahl
ETH Zürich::00002 - ETH Zürich::00012 - Lehre und Forschung::00007 - Departemente::02000 - Dep. Mathematik / Dep. of Mathematics::02003 - Mathematik Selbständige Professuren::03500 - Lang, Urs / Lang, Urs
en_US
ethz.date.deposited
2020-02-12T16:28:53Z
ethz.source
FORM
ethz.eth
yes
en_US
ethz.availability
Open access
en_US
ethz.rosetta.installDate
2020-02-13T10:26:48Z
ethz.rosetta.lastUpdated
2021-02-15T08:05:33Z
ethz.rosetta.versionExported
true
ethz.COinS
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