Thread: contractibility of convex subsets of R^n

The exercise: If A is a convex subset of R^n, show that A is contractible.

What I done: Fixing any x_0 in A, let f(x,t) := (1-t)x+t*x_0. f is continuous and for any x_1, f(x_1, t) is the line from x_1 to x_0 in A (i.e., is entirely inside of A by convexivity...convexness?...convexitude?...convale scence?), so f witnesses A to be contractible.

Is that it?

Originally Posted by cribby

The exercise: If A is a convex subset of R^n, show that A is contractible.

What I done: Fixing any x_0 in A, let f(x,t) := (1-t)x+t*x_0. f is continuous and for any x_1, f(x_1, t) is the line from x_1 to x_0 in A (i.e., is entirely inside of A by convexivity...convexness?...convexitude?...convale scence?), so f witnesses A to be contractible.

Is that it?

As boringly simple as that. Of course, your teacher may want you to explicitly build the homotopy of the identity map to a single point, but I think your thing captures the gist of the matter.

Tonio

Ps. I think it is "convexity"...but english is only my third language, so don't take me too seriously in this.

Great, thanks! "Homotopy", per se, has not entered our course yet so I think I'll leave it at that...although this wouldn't be the first or second or third time this instructor has provided exercises pertaining to concepts weeks away from being introduced. I'll see what the red ink says when I get my work back!

Originally Posted by tonio

As boringly simple as that. Of course, your teacher may want you to explicitly build the homotopy of the identity map to a single point, but I think your thing captures the gist of the matter.

Tonio

Ps. I think it is "convexity"...but english is only my third language, so don't take me too seriously in this.