  The first main result presented in the paper shows that the perimeters of
inner parallel sets of planar shapes having a given constant width are minimal
for the Reuleaux triangles. This implies that the areas of inner parallel sets
and, consequently, the inverse of the Cheeger constant are also minimal for the
Reuleaux triangles. Proofs use elementary geometry arguments and are based on
direct comparisons between general constant width shapes and the Reuleaux
triangle.
