No, this equation is false for non-convex polyhedra. Take a cube and remove from inside of it a smaller cube. The resulting body has the same average width (caliper diameter), but the sum of angles times the edges is different (there are negative summands coming).

If one wants a counterexample where the body is homeomorphic to a ball, then one can choose the inner cube to be removed very close to the boundary of the outer cube and make a short and narrow tunnel from the cavity to the outside. The tunnel contribution to the sum is small, so the sum is still different from that for the solid cube.

The average width can be extended to unions of convex bodies by a sort of inclusion-exclusion formula. For this, see for example Section 5 of

Klain, Daniel A.; Rota, Gian-Carlo, Introduction to geometric probability, Lezioni Lincee. Cambridge: Cambridge University Press. Rome: Accademia Nazionale dei Lincei, xiv, 178 p. (1997). ZBL0896.60004.

No, this equation is false for non-convex polyhedra. Take a cube and remove from inside of it a smaller cube. The resulting body has the same mean width (caliper diameter), but the sum of angles times the edges is different (there are negative summands coming).

If one wants a counterexample where the body is homeomorphic to a ball, then one can choose the inner cube to be removed very close to the boundary of the outer cube and make a short and narrow tunnel from the cavity to the outside. The tunnel contribution to the sum is small, so the sum is still different from that for the solid cube.

The mean width can be extended to unions of convex bodies by a sort of inclusion-exclusion formula. For this, see for example Section 5 of

Klain, Daniel A.; Rota, Gian-Carlo, Introduction to geometric probability, Lezioni Lincee. Cambridge: Cambridge University Press. Rome: Accademia Nazionale dei Lincei, xiv, 178 p. (1997). ZBL0896.60004.