The vertex index of centrally symmetric hexagons

The vertex index of an origin-symmetric planar convex body minimizes the sum of the gauges of the actual vertices of a finite enclosing polygon. Every centrally symmetric hexagon is shown to have an optimal enclosing quadrilateral centered at the origin. The sharp maximum within this class is 3 + 2 sqrt(2), attained exactly by affine-regular hexagons. Together with a previously constructed octagon, this identifies eight as the smallest number of vertices of a centrally symmetric polygonal body for which restricting enclosures to centered quadrilaterals raises the optimum. The proof reduces a hypothetical cheaper enclosure to six contact types; signed-parameter concavity and a harmonic-mean comparison dispose of the noncentral residual quadrilaterals. The principal results retain candidate status under AIM-R v2.6 Accepted. Editable sources, exact checks, supporting certificates and a Verification and Reproducibility Report accompany the article.