Geometry of Minkowski planes and spaces [Elektronische Ressource] : selected topics / vorglegt von Senlin Wu
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Geometry of Minkowski planes and spaces [Elektronische Ressource] : selected topics / vorglegt von Senlin Wu

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Geometry of Minkowski Planesand Spaces– Selected TopicsD I S S E R T A T I O Nzur Erlangung des akademischen GradesDoctor rerum naturalium(Dr. rer. nat.)Vorgelegt von M. Sc. Senlin Wugeboren am 16.01.1982 in ShanXi Provinz, VR ChinaGutachter: Prof. Dr. Horst Martini (TU Chemnitz)Prof. Dr. Gunter Weiß(TU Dresden)Prof. Dr. Eike Hertel (Friedrich-Schiller-Universitaet Jena)Tag der Verteidigung: 29. 01. 20091AbstractThe results presented in this dissertation refer to the geometry of Minkowskispaces, i.e., of real finite-dimensional Banach spaces.First we study geometric properties of radial projections of bisectors inMinkowski spaces, especially the relation between the geometric structure ofradial projections and Birkhoff orthogonality. As an application of our resultsit is shown that for any Minkowski space there exists a number, which plays√somehow the role that 2 plays in Euclidean space. This number is referredto as the critical number of any Minkowski space. Lower and upper bounds onthe critical number are given, and the cases when these bounds are attained arecharacterized. Moreover, with the help of the properties of bisectors we showthat a linear map from a normed linear space X to anothernormed linear spaceY preservesisoscelesorthogonalityif andonlyif itisascalarmultiple ofalinearisometry.

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Published 01 January 2009
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Language English

GeometryofMinkowskiPlanes
andSpaces
–SelectedTopics

DISSERTATION

zurErlangungdesakademischenGrades
Doctorrerumnaturalium
(Dr.rer.nat.)

gVeobrogreelengatmvo1n6.M01..S1c9.82SiennliSnhaWnuXiProvinz,VRChina

Gutachter:Prof.Dr.HorstMartini(TUChemnitz)
Prof.Dr.GunterWeiß(TUDresden)
Prof.Dr.EikeHertel(Friedrich-Schiller-UniversitaetJena)

TagderVerteidigung:29.01.2009

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Abstract

TheresultspresentedinthisdissertationrefertothegeometryofMinkowski
spaces,i.e.,ofrealnite-dimensionalBanachspaces.
Firstwestudygeometricpropertiesofradialprojectionsofbisectorsin
Minkowskispaces,especiallytherelationbetweenthegeometricstructureof
radialprojectionsandBirkhoorthogonality.Asanapplicationofourresults
itisshownthatforanpyMinkowskispacethereexistsanumber,whichplays
somehowtherolethat2playsinEuclideanspace.Thisnumberisreferred
toasthecriticalnumberofanyMinkowskispace.Lowerandupperboundson
thecriticalnumberaregiven,andthecaseswhentheseboundsareattainedare
characterized.Moreover,withthehelpofthepropertiesofbisectorsweshow
thatalinearmapfromanormedlinearspaceXtoanothernormedlinearspace
Ypreservesisoscelesorthogonalityifandonlyifitisascalarmultipleofalinear
isometry.
Furtheron,weexaminethetwotangentsegmentsfromanyexteriorpointto
theunitcircle,therelationbetweenthelengthofachordoftheunitcircleand
thelengthofthearccorrespondingtoit,thedistancesfromthenormalizationof
thesumoftwounitvectorstothosetwovectors,andtheextensionofthenotions
oforthocentricsystemsandorthocentersinEuclideanplaneintoMinkowski
spaces.AlsoweprovetheoremsreferringtochordsofMinkowskicirclesand
ballswhichareeitherconcurrentorparallel.Allthesediscussionsyieldmany
interestingcharacterizationsoftheEuclideanspacesamongall(strictlyconvex)
Minkowskispaces.
Inthenalchapterweinvestigatetherelationbetweenthelengthofaclosed
curveandthelengthofitsmidpointcurveaswellasthelengthofitsimage
undertheso-calledhalvingpairtransformation.Weshowthattheimagecurve
underthehalvingpairtransformationisconvexprovidedtheoriginalcurveis
convex.Moreover,weobtainseveralinequalitiestoshowtherelationbetween
thehalvingdistanceandotherquantitieswellknowninconvexgeometry.Itis
knownthatthelowerboundforthegeometricdilationofrectiablesimpleclosed
curvesintheEuclideanplaneis/2,whichcanbeattainedonlybycircles.We
extendthisresulttoMinkowskiplanesbyprovingthatthelowerboundforthe
geometricdilationofrectiablesimpleclosedcurvesinaMinkowskiplaneX
isanalogouslyaquarterofthecircumferenceoftheunitcircleSXofX,but
canalsobeattainedbycurvesthatarenotMinkowskiancircles.Inaddition
weshowthatthelowerboundisattainedonlybyMinkowskiancirclesifthe

2

respectivenormisstrictlyconvex.Alsowegiveasucientconditionforthe
geometricdilationofaclosedconvexcurvetobelargerthanaquarterofthe
perimeteroftheunitcircle.

Keywords:arclength,Birkhoorthogonality,bisectors,Busemannangu-
larbisector,C-orthocenter,characterizationsofEuclideanplanes,characteriza-
tionsofinnerproductspaces,chordlength,circumradius,convexcurve,convex
geometry,criticalnumber,detour,Euclideanplane,geometricdilation,geomet-
ricinequality,Glogovskijangularbisector,halvingdistance,halvingpair,inner
productspace,inradius,isometry,isoscelesorthogonality,Jamesorthogonality,
midpointcurve,minimumwidth,Minkowskiplane,Minkowskiplane,normed
linearspace,normedplane,radialprojection,Radonplane,rectication,Singer
orthogonality,strictlyconvexnorm,three-circlestheorem,Voronoidiagram.

3

Acknowledgment

Iwouldliketothankmyadvisor,Prof.Dr.HorstMartini,forhishelpful
support,usefulhints,fruitfulcollaboration(Theorem3.4.1,theideaofTheorem
4.2.2,proofideaswithrespecttoLemmas5.2.8and5.2.9,Theorem6.2.1,and
Lemma6.3.9inthisdissertationareduetohim),andtheseriesofhissurvey
paperswithProf.Dr.KonradJ.SwaneopoelandProf.Dr.GunterWeiß,
whichledmetotheresearcheldofMinkowskigeometry.
Prof.Dr.KonradJ.SwaneopoelandProf.Dr.JavierAlonsodeservespecial
creditforprovidingmewithusefulliteraturehints,encouragement,andgeneral
helpfulhintsevenwhenIwasstillworkingonmymaster’sthesis.Withoutthe
helpofProf.Dr.JavierAlonsoIwouldnotbefamiliarwiththeideasand
techniquestotreatproblemsrelatedtoorthogonalities.Itisagreatpleasure
toacknowledgeDr.MargaritaSpirovawhopointedoutamistakeintherst
versionoftheproofofTheorem5.4.7,andDr.WalterWenzelwhohelpedme
tondoutcertainreferencestosomeresultsinnumbertheory.
IamespeciallythankfultoProf.Dr.DonghaiJiwhowastheadvisor
ofmymaster’sthesisandledmetoproblemsconcerningorthogonalitiesand
MinkowskiGeometry.Healsohelpedmetondtheinitial