Local affine-isometry chart on a smaller ball

Set ; this is the radius used throughout the construction. In Radial-map definition, define and, for ,

The evaluation point is at distance from , hence belongs to .

For , the evaluation point for equals that for , and the prefactor is multiplied by . Thus Positive homogeneity of the radial map gives . Distance preservation gives Norm of the radial map in the concrete form

The case follows from the definition.

If , first write the point as the explicit convex combination

The coefficient lies in , and both endpoints lie in because . Apply Quarter-ball segment theorem to these two endpoints. It gives

The equality also holds for . This is Local agreement with the translated isometry.

For , their midpoint also has norm less than . Substitute the local agreement into Quarter-ball midpoint theorem for to obtain

This verifies the local identity in Local midpoint identity for the radial map. For arbitrary , choose . Both lie in . Apply the local identity to these two vectors and use positive homogeneity for the same :

Since , division by gives the midpoint identity for the original . This is the application of Rescaling local midpoint identities that proves the global midpoint identity.

Putting in the midpoint identity gives . It follows that . Consequently

Thus is continuous and additive, hence real-linear by rational approximation. The same midpoint map as a linear isometry equips this same function with the linear-isometry structure; it does not replace it by a different map.

Let . The target ball inclusion puts in . Take . Isometry gives , so local agreement yields

Hence . This range is a real vector subspace. For arbitrary , the scalar puts in that ball; if , then . Thus is surjective, without any dimension argument.

Define . Real linearity, norm preservation and surjectivity of the same make a surjective affine isometry. When , substitute into local agreement to obtain .