Hi to all! I have a question, i'm trying to prove that every star-shaped open subset of R^n is diffeomorphic to R^n. I found this sketch of proof:
Let U be an open and star-shaped region of R^n. Wlog let 0 be a star-center of U. There is a smooth function f:R^n -> R such that f>0 on U and f=0 outside U. Such a function certanly exists for open balls, and any open set of R^n is a locally finite union of open balls, so one just has to sum up all the funcion of the open balls in such a covering. Define g:U -> R by g(x) = integral_0^1 dt/f(tx). Define h:U -> R^n by h(x) = g(x)x. One can prove the following: 1 is one-to-one 2. h is onto 3. h is smooth 4. the Jacobian of h is invertible Then h is a smooth diffeomorphism by the inverse function theorem
how can i prove 1,2,4?
If anyone knows another way to prove that every open star convex subset of R^n is diffeomorphic to R^n tell me please.
Italo <shinr...@tin.it> wrote: > Hi to all! > I have a question, i'm trying to prove that every > star-shaped open subset of R^n is diffeomorphic to > R^n. > I found this sketch of proof:
> Let U be an open and star-shaped region of R^n. Wlog let 0 be a > star-center of U. There is a smooth function f:R^n -> R such > that f>0 on U and f=0 outside U. > Such a function certanly exists for open balls, and any open set > of R^n is a locally finite union of open balls, so one just has > to sum up all the funcion of the open balls in such a covering. > Define g:U -> R by g(x) = integral_0^1 dt/f(tx). > Define h:U -> R^n by h(x) = g(x)x. > One can prove the following: > 1 is one-to-one > 2. h is onto > 3. h is smooth > 4. the Jacobian of h is invertible > Then h is a smooth diffeomorphism by the inverse function theorem
> how can i prove 1,2,4?
For each unit vector v, let R(v) = sup {rv : r >= 0, rv in U}. Because of the rapid vanishing of f at the boundary of U, for all v we have g(rv) -> oo as r -> R(v) from below. Thus h maps the U-ray [0, R(v))*v onto the full ray [0, oo)*v. This shows h is onto. To see h is 1-1, note |h(rv)| = int_0^r dt/f(tv), which implies |h(rv)| is strictly increasing as a function of r.
For the Jacobian, let D_j denote the jth partial derivative and let e_j be the usual basis vector. Then D_jh(x) = g(x)*e_j + D_jg(x)*x. Thus the linear transformation Dh(x) equals g(x)*I + T_x, where I is the indentity and T_x is a linear transformation whose range is contained in the span of the vector x. Clearly Dh(0) equals g(0)*I, which is invertible. If x is nonzero, then the formula for |h(rv)| shows Dh(x) applied to x itself is a positive multiple of x. If w is a vector perpendicular to x, then Dh(x)(w) = g(x)*w + scalar multiple of x. It follows that the range of Dh(x) contains x and its orthogonal complement, which implies Dh(x) is invertible.
> Hi to all! > I have a question, i'm trying to prove that every > star-shaped open subset of R^n is diffeomorphic to > R^n. > I found this sketch of proof:
> Let U be an open and star-shaped region of R^n. Wlog > let 0 be a > star-center of U. There is a smooth function f:R^n -> > R such > that f>0 on U and f=0 outside U. > Such a function certanly exists for open balls, and > any open set > of R^n is a locally finite union of open balls, so > one just has > to sum up all the funcion of the open balls in such a > covering. > Define g:U -> R by g(x) = integral_0^1 dt/f(tx). > Define h:U -> R^n by h(x) = g(x)x. > One can prove the following: > 1 is one-to-one > 2. h is onto > 3. h is smooth > 4. the Jacobian of h is invertible > Then h is a smooth diffeomorphism by the inverse > function theorem
> how can i prove 1,2,4?
> If anyone knows another way to prove that every open > star convex subset > of R^n is diffeomorphic to R^n tell me please.
> Thank you in advance
I have a different idea that I think may work: We can show that every set of this type is diffeo- morphic to an open metric ball in R^n , i.e., a ball B={ x in R^n, ||x||<r } , for r>0. You can ( in the case the set is bounded ) contain the star set S in an open n-ball centered at the origin, and then draw a line from the origin passing thru a point in S and ending up at a point in the n-ball. This gives you a smooth bijection between the n-ball and S . And then it is relatively easy to show that an open metric ball in R^n is diffeomorphic to R^n. Use, e.g., some variant of the arctan function used for R^1.
Bacle <ba...@yahoo.com> wrote: > > Hi to all! > > I have a question, i'm trying to prove that every > > star-shaped open subset of R^n is diffeomorphic to > > R^n. > > I found this sketch of proof:
> > Let U be an open and star-shaped region of R^n. Wlog > > let 0 be a > > star-center of U. There is a smooth function f:R^n -> > > R such > > that f>0 on U and f=0 outside U. > > Such a function certanly exists for open balls, and > > any open set > > of R^n is a locally finite union of open balls, so > > one just has > > to sum up all the funcion of the open balls in such a > > covering. > > Define g:U -> R by g(x) = integral_0^1 dt/f(tx). > > Define h:U -> R^n by h(x) = g(x)x. > > One can prove the following: > > 1 is one-to-one > > 2. h is onto > > 3. h is smooth > > 4. the Jacobian of h is invertible > > Then h is a smooth diffeomorphism by the inverse > > function theorem
> > how can i prove 1,2,4?
> > If anyone knows another way to prove that every open > > star convex subset > > of R^n is diffeomorphic to R^n tell me please.
> > Thank you in advance
> I have a different idea that I think may work: > We can show that every set of this type is diffeo- > morphic to an open metric ball in R^n , i.e., > a ball B={ x in R^n, ||x||<r } , for r>0. > You can ( in the case the set is bounded ) > contain the star set S in an open n-ball centered > at the origin, and then draw a line from the > origin passing thru a point in S and ending up > at a point in the n-ball. This gives you a smooth > bijection between the n-ball and S .
That looks doubtful to me. Can you supply some details?
> And then it is relatively easy to show that > an open metric ball in R^n is diffeomorphic to > R^n. Use, e.g., some variant of the arctan > function used for R^1.
