Self-perimeter

The length $2\pi_S$ of a centrally-symmetric closed convex curve $S$ on a plane, measured in the Minkowski metric for which $S$ plays the role of unit circle. Always $3\leq\pi_S\leq4$ (see [1]). In the generalization to the non-symmetric case, a self-perimeter is the length of a directed curve $S$, and depends on the choice of the origin inside $S$ (see [2], [3]). The case of a star $S$ has been considered (see [4]). There are various generalizations of a self-perimeter for the unit sphere $S$ in a normed space of dimension greater than two (see [5], [6]).

References

[1]	Yu.G. Reshetnyak, "An extremal problem in the theory of surfaces" Uspekhi Mat. Nauk , 8 : 6 (1953) pp. 125–126 (In Russian)
[2]	V.A. Sorokin, "Minkowski geometry with asymmetric indicatrix" Uchebn. Zap. Moskov. Gos. Ped. Inst. , 243 (1965) pp. 160–185 (In Russian)
[3]	G.D. Chakerian, W.K. Talley, "Some properties of the self-circumference of convex sets" Arch. Math. , 20 : 4 (1969) pp. 431–443
[4]	S. Gołab, "Sur la longueur d'indicatrice dans la géométrie plane de Minkowski" Colloq. Math. , 15 : 1 (1966) pp. 141–144
[5]	J.J. Schäffer, "Spheres with maximum inner diameter" Math. Ann. , 190 : 3 (1971) pp. 242–247
[6]	C.M. Petty, "Geominimal surface area" Geom. Dedic. , 3 : 1 (1974) pp. 77–97