star shaped open sets
Italo
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
W^3
<569058.72611.12467132...@nitrogen.mathforum.org>,
Italo <shin...@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.
Italo
Bacle
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.
Think this should help.
join
W^3
<17069003.86560.1246990...@nitrogen.mathforum.org>
,
Bacle <ba...@yahoo.com> wrote:
That looks doubtful to me. Can you supply some details?