The vertex index of the affine-regular hexagon

The vertex index of an origin-symmetric planar convex body minimizes the sum of the gauges of the vertices of an enclosing polygon. This preprint gives an analytic proof that the affine-regular hexagon has vertex index 3 + 2√2. The infimum among enclosures with five actual vertices is 6, whereas a family of noncentral quadrilaterals attains the unrestricted minimum. A vertex-slide lemma reduces the problem to supporting polygons, and a complete contact-pattern analysis yields the lower bounds. The hexagon value corrects an asserted equality in the vertex-index literature. With Asplund’s equality condition and compactness, it also gives a non-explicit uniform planar upper bound strictly below 6. The principal results retain candidate status; no sharp universal constant or classification of all minimizers is claimed.