Unit disk of the smallest self-perimeter in a Minkowski plane (Q2460514)

Let \(B\) be a convex body in an affine plane \(A\) containing the origin \(O\) in its interior (not necessarily symmetric about \(O\)). Consider the Euclidean plane obtained from \(A\) by specifying a scalar product on \(A\). For distinct points \(X, Y\) in \(A\), define a distance \(d(X,Y)\) from \(X\) to \(Y\) as follows. Consider a ray from \(O\) in the direction of the vector \(Y-X\) and let \(Z\) be the point of its intersection with the boundary of \(B\). Put \(d(X,Y)= |Y-X|/|Z|\). Thus, \(d\) is a (non-symmetric) metric. Let \(P\) be a convex bounded polygon. Denote by \(l^+(P)\) its perimeter traversed counterclockwise and by \(l^-(P)\) this perimeter traversed clockwise. For a compact convex figure \(K\), define perimeters \(U^\pm (K) = \sup l^\pm (K)\) where the supremum is taken over all convex polygons \(P\) within \(K\). On the Minkowski plane with a norming figure \(B\) symmetric about \(O\), both perimeters have a common value \(U(K)\). Golab proved in 1932 that the ``self-perimeter'' \(U(B)\) satisfies \(6\leq U(B)\leq 8\). Schaeffer proved in 1967 that \(U(B)=6\) only if \(B\) is affinely regular hexagon centered at \(O\). For a not necessarily symmetric norming figure \(B^*\), the author proved earlier that \(U^\pm(B^*)\geq 6\). (There is no upper bound for \(U^\pm(B^*)\).) The main result of this paper is the following. Theorem. The equality \(U^-(B^*)=6\) or \(U^+(B^*)=6\) holds if and only if \(B^*\) is an affinely regular hexagon. (Hence, one equality implies the other one.)