Convex tessellation property of weighted straight skeletons

Prove that the weighted straight skeleton construction used to partition a convex polygon into regions associated with its edges always yields a convex tessellation under the stated general-position assumptions.

Background

The paper's second subdivision tool is essentially equivalent to computing a weighted straight skeleton. The resulting regions are used to obtain a convex partition with prescribed visibility properties.

Although the authors establish convexity directly for their specific construction, they cannot locate a proof in the literature that the general weighted straight skeleton produces a convex tessellation. Establishing this property would support the proposed generalization of the subdivision method.