We prove a stability result on the minimal self-perimeter L(B) of the unit disk B of a normed plane: if L(B) = 6 + ε for a sufficiently small ε, then there exists an affinely regular hexagon S such that S ⊂ B ⊂ (1 + 6∛ε) S.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm131-1-7,
author = {Horst Martini and Anatoly Shcherba},
title = {On the stability of the unit circle with minimal self-perimeter in normed planes},
journal = {Colloquium Mathematicae},
volume = {131},
year = {2013},
pages = {69-87},
zbl = {1285.46010},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm131-1-7}
}
Horst Martini; Anatoly Shcherba. On the stability of the unit circle with minimal self-perimeter in normed planes. Colloquium Mathematicae, Tome 131 (2013) pp. 69-87. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm131-1-7/