LAUSR

We study the pth power $$W_p(E)=\int _{{\mathbb R}^n}|H_E|^p\,dx$$Wp(E)=∫Rn|HE|pdx, where $$H_E$$HE is the variational mean curvature of E, as defined in Barozzi (Rend Mat Acc Linceis 9(5):149–159, 1994) (see also in Proc Am Math Soc 99:313–316, 1987). This defines a functional $$E\rightarrow W_p(E)$$E→Wp(E) on all bounded sets of finite perimeter. We prove that this functional is lower semicontinuous for every $$p\ge 1$$p≥1, with respect to the $$L^1({\mathbb R}^n)$$L1(Rn) topology. A corresponding compactness theorem also holds. The case $$p=n$$p=n appears to be particularly interesting. Finally, we introduce the pseudoconvex hull K of a bounded set $$E\subset {\mathbb R}^n$$E⊂Rn and establish the inequality $$W_p(K)\le W_p(E)$$Wp(K)≤Wp(E).