Contractible metric space
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From: Julien Santini on 25 Feb 2005 14:53 Hello, I have a problem of terminology. What is meant by "starlike" in the assertion: "every starlike metric space is contractible" ? The definition given by the book is: "A metric space M with metric d is starlike in that metric if there is a point p in M such that each other point x in M can be joined to p by a unique arc congruent in the metric of M to a line segment". What is meant by "unique" ? What is meant by arc ? (traditionally an arc is a continuous map defined on [0,1] right ?) And what about "congruent [...] to a line segment ?" I would appreciate if someone could translate this definition using a mathematical formalism only. Thank you -- Julien Santini
From: Stuart M Newberger on 25 Feb 2005 17:17 Julien Santini wrote: > Hello, > > I have a problem of terminology. What is meant by "starlike" in the > assertion: "every starlike metric space is contractible" ? > > The definition given by the book is: "A metric space M with metric d is > starlike in that metric if there is a point p in M such that each other > point x in M can be joined to p by a unique arc congruent in the metric of M > to a line segment". > > What is meant by "unique" ? What is meant by arc ? (traditionally an arc is > a continuous map defined on [0,1] right ?) And what about "congruent [...] > to a line segment ?" > > I would appreciate if someone could translate this definition using a > mathematical formalism only. > > Thank you > > -- > Julien Santini Here is my guess."Arc congruent in the metric M means an isometry from [0,1] (usual metric) into M. So for each q in M there is a unique map f=f_q :[0,1]->M with f(0)=p and f(1)=q and d(f(x),f(y))=|x-y| for each x,y in [0,1].Try the problem now. Regards,Stuart M Newberger
From: Timothy Little on 25 Feb 2005 17:49 Julien Santini wrote: > What is meant by "unique"? There is only one object satisfying the properties. Formally, if we let P(x) be a predicate corresponding to the properties, then P(A) and P(B) => A = B. > What is meant by arc ? (traditionally an arc is a continuous map > defined on [0,1] right ?) Almost: that's a path. An arc is more restricted: its function must be bijective and continuous in both directions. In mathematical formalism: f:[0,1] -> S is an arc in M iff S is a subspace of M and f is bijective, continuous, and has continuous inverse. > And what about "congruent [...] to a line segment ?" There exists an distance preserving bijection between an interval of R (i.e. a line segment) and the arc. Formally: A subspace S of M is congruent to a line segment iff there exists a,b in R and f:[a,b] -> S such that f is bijective and for all x,y in [a,b], d(f(x),f(y)) = |x-y|. - Tim
From: Michael Barr on 25 Feb 2005 21:02 "Julien Santini" <santini.julien(a)wanadoo.fr> wrote in message news:<421f8253$0$19361$8fcfb975(a)news.wanadoo.fr>... > Hello, > > I have a problem of terminology. What is meant by "starlike" in the > assertion: "every starlike metric space is contractible" ? > > The definition given by the book is: "A metric space M with metric d is > starlike in that metric if there is a point p in M such that each other > point x in M can be joined to p by a unique arc congruent in the metric of M > to a line segment". > > What is meant by "unique" ? What is meant by arc ? (traditionally an arc is > a continuous map defined on [0,1] right ?) And what about "congruent [...] > to a line segment ?" > > I would appreciate if someone could translate this definition using a > mathematical formalism only. > > Thank you I will take a guess at this. Unique means unique (what else could it mean). I take it that congruent means isometric. That is there is a point * such that for each other point x, there is one and only one map u_x: [0,1] --> X s.t. u(0) = *, u_x(1) = x and dist(u_x(s),u_x(t)) = |s-t| for any s,t in [0,1]. I guess this allows you to prove that the function h: X x [0,1] --> X defined by h(x,t) = u_x(t) contracts X to *.
From: Julien Santini on 26 Feb 2005 08:58
> I will take a guess at this. Unique means unique (what else could it > mean). I take it that congruent means isometric. That is there is a > point * such that for each other point x, there is one and only one > map u_x: [0,1] --> X s.t. u(0) = *, u_x(1) = x and dist(u_x(s),u_x(t)) > = |s-t| for any s,t in [0,1]. I guess this allows you to prove that > the function h: X x [0,1] --> X defined by h(x,t) = u_x(t) contracts X > to *. That's what I had thought, but my problem is that although I can show h_x: [0,1] -> X, h_x(t) = h(x,t) is continuous (using this definition of starlike), I cannot show h_t: X->X, h_t(x) = h(x,t) is continuous. Working in R^2 for instance, and considering an arc (*p), with a point x on this arc, and a neighborhood U of x, why couldn't U intersect (no matter how "small" U is) another arc at point y so that for some fixed t, and for some fixed e>0, abs(h_t(y)-h_t(x))>e (i.e h_t is not continuous) ? That's what bothers me. -- Julien Santini |