  For certain families of compact subsets of the plane, the isomorphism class
of the algebra of absolutely continuous functions on a set is completely
determined by the homeomorphism class of the set. This is analogous to the
Gelfand--Kolmogorov theorem for $C(K)$ spaces. In this paper we define a family
of compact sets comprising finite unions of convex curves and show that this
family has the `Gelfand--Kolmogorov' property.
