Its interior is the open -ball
More generally, for a metric space then an open ball in is a subset of the form
A simple result on the homeomorphism type of closed balls is the following:
Without loss of generality we may suppose the origin is an interior point of . We claim that the map maps the boundary homeomorphically onto . By convexity, is homeomorphic to the cone on , and therefore to the cone on which is .
The claim reduces to the following three steps.
The restricted map is continuous.
It’s surjective: contains a ball in its interior, and for each , the positive ray through intersects in a bounded half-open line segment. For the extreme point on this line segment, . Thus every unit vector is of the form for some extreme point , and such extreme points lie in .
It’s injective: for this we need to show that if are distinct points, then neither is a positive multiple of the other. Supposing otherwise, we have for , say. Let be a ball inside containing ; then the convex hull of is contained in and contains as an interior point, contradiction.
So the unit vector map, being a continuous bijection between compact Hausdorff spaces, is a homeomorphism.
Any compact convex set of is homeomorphic to a disk.
has nonempty interior relative to its affine span which is some -plane, and then is homeomorphic to by the theorem.
Open balls are a little less rigid than closed balls, in that one can more easily manipulate them within the smooth category:
For instance, the smooth map
has smooth inverse
In dimension for we have:
See the first page of (Ozols) for a list of references.
This is a folk theorem. But explicit proofs in the literature are very hard to find. See the discussion at References. An explicit proof has been written out by Stefan Born, and this appears as the proof of theorem 237 in (Ferus).
Suppose is a star-shaped open subset of centered at the origin. Theorem 2.29 in Lee proves that there is a function on such that on and vanishes on the complement of . By applying bump functions we can assume that everywhere and in an open -neighborhood of the origin; by rescaling the ambient space we can assume .
The smooth vector field is defined on the complement of the origin in . Multiply by a smooth bump function such that for and in a neighborhood of 0. The new vector field extends smoothly to the origin and defines a smooth global flow . (The parameter of the flow is all of and not just some interval because the norm of is bounded by 1.) Observe that for the vector field equals . Also, all flow lines of are radial rays.
Now define the flow map as for . (The subscript removes the closed ball of radius .) The flow map is the composition of two diffeomorphisms,
hence itself is a diffeomorphism. (Note particularly that the latter map is surjective. In detail: a flow line is a smooth map of the form , where and can be finite or infinite. If is finite and the limit of as exists, then the vector field vanishes at . In our case can only vanish at the boundary of , which is precisely what we want for surjectivity.)
Finally, define the desired diffeomorphism as the gluing of the identity map for and as for . The map is smooth because for both definitions give the same value.
Parameterize the -simplex as
Then define the map by
(Thanks to Todd Trimble.) One way to think about it is that is the positive orthant of an open -ball in norm, so that in the opposite direction we have a chain of invertible maps
which we simply invert to get the map above.
That an open subset homeomorphic to equipped with the smooth structure inherited as an open submanifold of might nevertheless be non-diffeomorphic to , see
The proof that open star-shaped regions are diffeomorphic to a ball appears as theorem 237 in
It is a lengthy proof, due to Stefan Born.
A simpler version of the proof apparently appears on page 60 of
Apparently this proof is little known. For instance in a remark below lemma 10.5.5 of
It seems that open star shaped sets are always diffeomorphic to , but this is extremely difficult to prove.
one finds the statement:
Actually, the assertion that an open geodesically convex set in a Riemannian manifold is diffeomorphic to is common in literature, but it is a more subtle issue than it may seem, and references to a complete proof are hard to find (but see [Grom]).
Here “Grom” refers to
where the relevant statement is 1.4.C1 on page 8. Note however that the diffeomorphism considered there is only of class, not , so that this is not a proof, either.
For a discussion of diffeomorphisms between geodesically convex regions and open balls see good open cover.
See also the Math Overflow discussion here.