On the stability of the unit circle with minimal self-perimeter in normed planes (Q2839279)

Let \(M=(\mathbb R^2, \|\cdot\|)\) be a normed (or Minkowski) plane with unit disc \(B:=\{x\in \mathbb R^2: \|x\|\leq 1\}\). If \(P\subset \mathbb R^2\) is a convex polygon, let \(l(P)\) denote the sum of the lengths (with respect to the norm) of all sides of \(P\). For a convex body \(K\), i.e., a compact, convex set with non-empty interior, \(\{K\}\) is the set of all convex polygons which are inside of \(K\). The (norm) perimeter \(L(K)\) of \(K\) is defined by \( L(K):=\sup_{P\in \{K\}} l(P)\). The perimeter \(L(B)\) of the unit disc is called its self-perimeter. According to the classical results of \textit{S. Golab} [Colloq. Math. 15, 141--144 (1966; Zbl 0141.20002)] and \textit{J. J. Schäffer} [Math. Ann. 173, 59--79 (1967; Zbl 0152.12405)], we have NEWLINE\[NEWLINE6\leq L(B)\leq 8,NEWLINE\]NEWLINE with \(L(B)=6\) if and only if \(B\) is an affinely regular hexagon and \(L(B)=8\) if and only if \(B\) is a parallelogram. In the paper under review, it is proved that if \(L(B)=6+\varepsilon\) for a sufficiently small \(\varepsilon> 0\), then there exists an affinely regular hexagon \(S\) such that \(S\subset B\subset(1+6 \root3\of{\varepsilon})S\).