Before revision:  We prove that HCS satisfies some properties analogous to those of ACSF: HSC is invariant under affine transformations, preserves convexity, and does not increase the total absolute curvature.
After revision:  We prove that HCS satisfies some properties analogous to those of ACSF: HCS is invariant under affine transformations, preserves convexity, and does not increase the total absolute curvature.