Boundary Half-Strips and the Strong CHIP

Abstract

When the subdifferential sum rule formula holds for the indicator functions $\iota_C$ and $\iota_D$ of two closed convex sets C and D of a locally convex space X, the pair $(C,D)$ is said to have the strong conical hull intersection property (the strong CHIP). The specification of a well-known theorem due to Moreau to the case of the support functionals $\sigma_C$ and $\sigma_D$ subsumes the fact that the pair $(C,D)$ has the strong CHIP whenever the inf-convolution of $\sigma_C$ and $\sigma_D$ is exact. In this article we prove, in the setting of Euclidean spaces, that if the pair $(C,D)$ has the strong CHIP while the boundary of C does not contain any half-strip, then the inf-convolution of $\sigma_C$ and $\sigma_D$ is exact. Moreover, when the boundary of a closed and convex set C does contain a half-strip, it is possible to find a closed and convex set D such that the pair $(C,D)$ has the strong CHIP while the inf-convolution of $\sigma_C$ and $\sigma_D$ is not exact. The validity of the converse of Moreau's theorem in Euclidean spaces is thus associated with the absence of half-strips within the boundary of concerned convex sets.