Smooth maps on convex sets

TL;DR

The paper investigates different notions of smoothness on convex sets, demonstrating their differences and coincidences, and establishing an isomorphism between diffeological spaces and exhaustive Chen spaces.

Contribution

It constructs a specific example illustrating the divergence of smoothness notions and proves their equivalence under local closedness, linking diffeological and Chen spaces.

Findings

Certain smoothness notions do not coincide on convex sets.

On locally closed convex sets, these notions are equivalent.

The category of diffeological spaces is isomorphic to exhaustive Chen spaces.

Abstract

There are several notions of a smooth map from a convex set to a cartesian space. Some of these notions coincide, but not all of them do. We construct a real-valued function on a convex subset of the plane that does not extend to a smooth function on any open neighbourhood of the convex set, but that for each $k$ extends to a $C^k$ function on an open neighbourhood of the convex set. It follows that the diffeological and Sikorski notions of smoothness on convex sets do not coincide. We show that, for a convex set that is locally closed, these notions do coincide. With the diffeological notion of smoothness for convex sets, we then show that the category of diffeological spaces is isomorphic to the category of so-called exhaustive Chen spaces.