Symmetry breaking in the planar vertex index

The vertex index of an origin-symmetric convex body minimizes the sum of the gauges of the vertices of a finite enclosing polytope. A rational planar octagon is exhibited for which an asymmetric quadrilateral is cheaper than every centrally symmetric finite enclosing polygon, even when its center may move. The quadrilateral costs 49115/9016, whereas the exact centered minimum is (134 + 4 sqrt(1110))/49. The candidate centered lower bound has a computer-assisted proof: polarity reduces all centered quadrilaterals to ten parameter domains, and a complete exact cell calculation leaves one geometric edge whose minimum follows from a square identity. An explicit l_8192 construction extends the strict separation to a smooth strictly convex body. Exact unrestricted vertex-index values are not determined.