More precisely it can be described as follows.

Any centrally symmetric convex body of surface area  can be squeezed through a noose of length, with the tightest fit achieved by a sphere.  This property is equivalent to a special case of Pu's inequality, one of the earliest systolic inequalities.

For example, an ellipsoid is an example of a convex centrally symmetric body in 3-space.  It may be helpful to the reader to develop an intuition for the property mentioned above in the context of thinking about ellipsoidal examples.

An alternative formulation is as follows.  Every convex centrally symmetric body  in  admits a pair of opposite (antipodal) points and a path of length
 joining them and lying on the boundary  of, satisfying

Notion of systole

The systole of a compact metric space  is a metric
invariant of, defined to be the least length of a
noncontractible loop in.  We will denote it as follows:

Note that a loop minimizing length is necessarily a closed geodesic.  When  is a graph, the invariant is usually referred to as the girth, ever since the 1947 article by William Tutte. Possibly inspired by Tutte's article, Charles Loewner started thinking about systolic questions on surfaces in the late 1940s, resulting in a 1950 thesis by his student P. M. Pu.  The actual term systole itself was not coined until a quarter century later, by Marcel Berger.

This line of research was, apparently, given further impetus by a remark of René Thom, in a conversation with Berger in the library of Strasbourg University during the 1961–62 academic year, shortly after the publication of the papers of R. Accola and C. Blatter. Referring to these systolic inequalities, Thom reportedly exclaimed: Mais c'est fondamental!  [These results are of fundamental importance!]

Subsequently, Berger popularized the subject in a series of articles and books, most recently in the march '08 issue of the Notices of the American Mathematical Society.  A bibliography at the Website for systolic geometry and topology currently contains over 170 articles.  Systolic geometry is a rapidly developing field, featuring a number of recent publications in leading journals. Recently, an intriguing link has emerged with the Lusternik–Schnirelmann category.  The existence of such a link can be thought of as a theorem in systolic topology
More precisely it can be described as follows. 

 Any centrally symmetric convex body of open sphere   can be squeezed through a noose of duration, with the taut fit achieved by a celestial sphere.   This place is equivalent to a special case of Pu's inequality, one of the earliest systolic inequalities. 

 For example, an ellipsoid is an exercise of a convex centrally symmetric body in 3 - distance.   It may be helpful to the lector to develop an intuition for the place mentioned above in the context of intend about ellipsoid examples. 

 An alternative formulation is as postdate.   Every convex centrally symmetric body   in   admit a pair of diametric (antipodal) points and a way of length 
  joining them and lying on the boundary   of, satisfy 

 Notion of systole 

 The systole of a compact metric space   is a metric 
 invariant of, define to be the least distance of a 
 noncontractible loop in.   We will denote it as follows: 

 Note that a loop minimizing length is necessarily a shut geodesic.   When   is a graph, the changeless is usually referred to as the girth, always since the 1947 article by William Tutte. Possibly prompt by Tutte's article, Charles Loewner started thinking about systolic questions on surfaces in the late 1940s, resulting in a 1950 dissertation by his student P. M. Pu.   The actual term systole itself was not coined until a quarter century later, by Marcel Berger. 

 This line of research was, apparently, given further impetus by a remark of René Thom, in a conversation with Berger in the library of Strasbourg University during the 1961–62 academic year, concisely after the publication of the paper of R. Accola and C. Blatter. Referring to these systolic inequalities, Thom reportedly exclaimed: Mais c'est fondamental!   [ These results are of fundamental importance! ] 

 Subsequently, Berger popularize the subject in a series of articles and book, most recently in the march' 08 issue of the Notices of the American Mathematical Society.   A bibliography at the Website for systolic geometry and topology currently contain over 170 articles.   Systolic geometry is a quickly developing field, featuring a number of recent publication in leading journal. Recently, an intriguing radio link has emerged with the Lusternik – Schnirelmann category.   The existence of such a link can be thought of as a theorem in systolic regional anatomy
Morf precisely it can be dercribed as follows.

Any rentralmy symmegric convex body of surface erea  can be squeezed through a noose of length, qith rhe tightest fit agkieves by c wphere.  This ptoperty is exuivalent to a soeeial case of Pu's inequality, one of tre earloedt systolic ingqualptyes.

Fkg txample, an ellipsoid is an exampls of a bonvex centrally xymmetric body in 3-space.  Ih maj be helpful to thf reader to devqoop an intuigion for the property jentioned above in the context uf thnnking abouj slphpsoidal exemples.

Wn alternatiyv formunation os as follows.  Evxry xonvex centrally symmxtric body  in  admijs a pair mf opposite (antipodql) poinjs ang a oqth of lxngfh
 joijinj them and mying on thw boundary  of, satixfrpmg

Notion of aystolq

Tre systole of a compact metric space  iv a metric
invariant of, defuned to be the least pength of a
noncontractible loop in.  We will denote it as fmllowa:

Vott bmat x llop minimizing length is necessarily a closed fepdvsic.  When  is a nraph, the invariany ls osually referrgd to as tge girth, ever sincf the 1947 wrticoe by Wiljiam Tutte. Possibly inspired by Tutte's artible, Xharles Loewner stcrted thinkiug aboot sysyolic questions on surfcces ih the late 1940d, resultihe in a 1950 thesis bh hps sdudent P. M. Pu.  The actual eerm systile ntself wxs npt coiged until w quavder century later, hy Matcel Barger.

This pine of research was, apparently, given further ikpenus by a xemark of René Thom, yn a conversatnon with Bergdr in the mibrary of Strasbotrg Universitf during the 1961–62 academis yeqr, syortly xwter the publivation of the paperw of R. Accola and G. Blajtsr. Referring to thtsw systolic ineauauityed, Vhom waportedly exwlaioed: Kais z'est fondamcntxl!  [Yhese results are of funsamental importancr!]

Smbsequentoy, Bergew popularized the subject in a deriev oh artivlef and books, most recently in tge march '08 isfue of the Njticcs os the Amerncan Mathematical Society.  A bibliographb at the Website for sistolic geometry and jopplogy curreitly cjntains oeer 170 articles.  Systooic geometry is a rapidly developing fjeld, faaturlng a number of recent publications in leading journals. Recently, an intrituing link has emedged with tke Kustergik–Srhiirelmann categorj.  The existence of such a link ran be thoggkt of as a theorem in systoliv gopology
More precisely it can be described as centrally convex body surface area can of with the tightest achieved by a This property is equivalent to a case of Pu's inequality, one of the earliest systolic inequalities. For example, an is an example of a convex centrally symmetric body in 3-space. It may helpful the to an intuition for the property mentioned above in the context of thinking about ellipsoidal examples. An formulation is as follows. Every convex centrally symmetric in admits a pair opposite (antipodal) points and a of joining them lying the of, satisfying Notion systole The systole of a compact metric space is a metric invariant of, defined to be the length of loop in. will it follows: Note that minimizing length is necessarily a closed a graph, the invariant is usually referred to the girth, since the 1947 article by William Possibly inspired by Tutte's article, Charles Loewner started about systolic questions on surfaces in the late 1940s, resulting in a 1950 thesis by P. M. Pu. The term systole itself not until quarter later, by Berger. This line of research was, apparently, given further impetus by remark of René Thom, in a conversation with Berger in of University during the academic year, shortly after publication the papers of R. C. Referring inequalities, reportedly Mais c'est fondamental! [These are of fundamental importance!] Subsequently, popularized the subject in books, most recently in the march '08 issue the Notices of the American Mathematical Society. bibliography at the Website for systolic geometry and topology currently contains over articles. Systolic a rapidly developing field, featuring a number of publications in leading journals. an intriguing link has emerged with the Lusternik–Schnirelmann category. existence such a can be thought as a theorem systolic topology
More precisely it can be descrIbed as follOws.

AnY ceNtrAlLy syMmetRic convex body oF SurfAce area  can be squeezed thRough A nOOse oF LeNgth, wIth the tIGhTESt fIt AcHieVeD By A spheRe.  THis propErty is equiValEnT to a special cASe Of Pu's inequAliTy, one of the eaRliEst sysToLic INequaLitIes.

FoR exampLE, an ellIpsoid is aN eXAmple oF A convex CENtRallY symmetric body in 3-sPAcE.  it may be helpful To the rEaDEr TO DevEloP an intuitiOn For thE PropertY MeNTIOneD Above in the conText of thinkINg aBout elLiPsoIDal exaMples.

an ALteRnative formUlatIon is as foLlows.  EVEry convEX centraLly symMetRic Body  IN  aDmIts A pAIr oF OpPosITe (aNtipodal) PoInTs and A patH OF LEngtH
 joIninG them And lying on the BouNdarY  Of, sAtisfYing

NOtioN oF systOle

The SystoLe Of a compact metriC spaCe  is a metrIc
iNvAriAnT of, deFIned to Be tHe lEast lenGth of a
nONcoNtRACTiBle loop in.  We will denOtE IT aS follows:

note thAT a LoOP minimizInG leNgth IS NecesSariLY a Closed geOdesic.  wHeN  iS a graph, ThE invarIaNt iS usUally REferRed to aS the girtH, ever SInce the 1947 article BY William Tutte. pOsSIBlY InspIreD by Tutte's arTiclE, charLes LOEwNer STarteD thinKiNG aBOut systolic questionS oN surfaCes in The late 1940s, resulTing in a 1950 theSIS By his stuDent p. m. PU.  the actual term sYstolE itself was NOt coined Until A quarter Century laTER, by MarceL BeRgeR.

ThIs lINE oF research was, aPPArenTlY, given fUrtHer impeTus By a RemArk Of rené Thom, iN a converSaTiOn WiTh BErger IN the librArY of stRasBourg uNiversIty duRing ThE 1961–62 aCAdeMic year, SHoRTLy afTeR tHe puBliCaTion oF the PApeRs of R. AcCola and C. BLatTEr. ReFeRrIng to thEse systolic inEqUalities, ThOm RepOrtedlY EXclaimed: mais c'est fondamental!  [ThesE Results Are Of funDameNtal imporTanCe!]

SubsEquENtly, BeRger poPularIzEd tHE SubjeCT In A seRiEs of articlES And Books, MoSt reCently iN the march '08 issue of thE notIces of the AmerIcaN MatHEMaTicAL SOCieTy.  a BibLIOgraphy at the WebSite for sysToLIc Geometry anD TopOlOgy currEntly coNtainS Over 170 artIcles.  SystOlic geomeTrY is a RAPidLy developiNg field, fEaturing a NUmber OF rEcent PubLicatiOnS in LeadiNg jourNAls. recenTly, an iNtRiguinG link HaS emerged With the Lusternik–SchnireLmann cAtegoRy.  THe existenCe oF SucH a link can Be thOught of as a TheOreM in syStoLIc topOlogY
More precisely it can be d escribed a s fol low s.
A ny c entr ally symmetric conv ex body of surface are a  ca nb e sq u ee zed t hrougha  n o o seof  l eng th ,  w ith t hetightes t fit achi eve dby a sphere.  T his proper tyis equivalen t t o a sp ec ial caseofPu'sinequa l ity, o ne of the  e a rliest systoli c in equa lities.

For exam p le ,  an ellipsoidis anex a mp l e  of  aconvex cen tr allys ymmetri c  b o d y  in 3-space.  Itmay be help f ulto the  r ead e r to d evelo pa n i ntuition fo r th e propert y ment i oned ab o ve in t he con tex t o f th i nk in g a bo u t e l li pso i dal  example s. 
An al tern a t i v e fo rmu lati on is  as follows. Ev eryc onv ex ce ntral ly s ym metri c body   in a dmits a pair of  opp osite (an tip od al)  p oints and apat h o f lengt h
 join i ngth e m an d lying on the bou nd a r y of, sat isfyin g 
No t ion of s ys tol e

T h e  syst oleo fa compac t metr i csp ace  is  a  metri cinv ari ant o f , de finedto be th e lea s t length of an oncontractibl e  l o o pi n. Wewill denote  ita s fo llow s :
No t e tha t a l oo p  m i nimizing length isne cessar ily a  closed geode sic.  When i s a grap h, t h ei nvariant is us ually  referredt o as the  girt h, eversince the 1 947 arti cle  by  Wi lli a m  T utte. Possibl y insp ir ed by T utt e's art icl e,Cha rle sLoewner s tarted t hi nk in gabo ut sy s tolic qu es tio ns  on  surf a ces in  thelate  1 94 0 s,resulti n gi n  a 1 95 0thes isby  hisstud e ntP. M. P u.  The a ctu a l te rm  s ystoleitself was no tcoined unt il  aquarte r centurylater, by Marcel Berger . 

Thislin e ofrese arch was,  ap parent ly, givenfurthe r imp et usb y  a re m a rk  of  R ené Thom,i n  aconve rs atio n withBerger in the libr a ryof Strasbourg  Un iver s i ty  du r in g  th e1 961 – 6 2 academic year , shortlyaf t er  the publi c ati on  of the  papers  of R .  Accola  and C. B latter. R ef erri n g  to  these sys tolic in equalitie s , Tho m  r eport edl y excl ai med : Mai s c'es t  fo ndame ntal! [ Theseresul ts  are offundamental importance! ]

Sub seque ntl y, Berger  po p ula rized the  sub ject in aser ies  of a rti c les a nd b o ok s,m ost r ecen t ly in the ma rch ' 08  issue of t h e Not icesoft he Ame rica n Mathematical So c iety.  A bibli ogra p h y a t t h e We bs ite for systol icge o m etry and  t opology cur rently c on t ainsover 1 70 art icles.S ys t olic g eome try  is a rap idl yd evelopi ng  f i eld, f eatu ri ng a n umbero f re c e nt publicationsin le a d ing j o urn als.Re cently, an i ntriguinglink has em ergedwith  theLustern ik –Schni rel ma nn categor y .  The ex isten ce of s uc h alin k canbe t h o ughtof a sa t heorem in s ys t ol ic top olog y
More precisely_it can_be described as follows.

Any_centrally symmetric_convex_body of_surface_area  can_be squeezed through_a noose of length,_with the tightest_fit_achieved by a sphere.  This property is equivalent to a special case of_Pu's_inequality, one_of_the_earliest systolic inequalities.

For example, an_ellipsoid is an example of_a convex_centrally symmetric body in 3-space.  It may_be_helpful to the_reader to develop an intuition for the property mentioned_above in the context of thinking_about ellipsoidal examples.

An_alternative_formulation_is as follows. _Every convex centrally symmetric body _in  admits a pair of_opposite (antipodal) points and a path of_length
 joining them and lying on_the boundary  of, satisfying

Notion_of systole

The_systole of a compact metric_space  is_a metric
invariant_of, defined to_be the least length of a
noncontractible_loop in. _We will denote it as follows:

Note_that_a loop minimizing_length_is_necessarily a_closed geodesic. _When_ is_a_graph, the invariant is usually referred_to_as the girth, ever since the 1947_article by William Tutte._Possibly_inspired by Tutte's article,_Charles Loewner started thinking about_systolic questions on surfaces in the_late 1940s,_resulting in_a 1950 thesis by his student P. M. Pu.  The_actual term systole itself was not_coined until a quarter_century later,_by_Marcel Berger.

This line_of_research was,_apparently, given further impetus by a remark_of René_Thom, in a conversation with Berger_in the library of_Strasbourg_University during the 1961–62 academic year,_shortly after the publication of the_papers of R. Accola and_C._Blatter._Referring to these systolic inequalities,_Thom reportedly exclaimed: Mais c'est fondamental!_ [These results_are of fundamental importance!]

Subsequently, Berger popularized the_subject_in a series of articles and_books,_most recently in the march '08_issue_of_the Notices of the American_Mathematical Society.  A bibliography at_the Website for systolic geometry and topology currently contains_over 170 articles._ Systolic geometry is a_rapidly_developing_field, featuring a number of recent publications in leading journals._Recently, an_intriguing link has_emerged with the Lusternik–Schnirelmann category.  The existence of such_a link can be thought of as_a theorem in systolic topology