On the self-perimeter of quadrangles for gauges
Let \(B\) be a compact convex set of an affine plane \(A^2\) and let \(N\) be a point in \(Int(B)\). A plane \(A^2\) endowed with a gauge metric (possibly non-symmetric) derived from \(B\) and \(N\) is called a Minkowski plane, \(M^2\). The point \(N\) is said to be the origin of \(M^2\), while \(B\) is called the normalizing figure, or the unit ball, of \(M^2\). We thus have a well-defined Minkowski length of a segment and, consequently, a Minkowski length of any curve in \(M^2\). For a non-symmetric metric, this length is sensitive to orientation. If \(P\) is a convex polygon in \(M^2\), let \(l^+(P)\) denote the perimeter of \(P\) measured in the counter-clockwise orientation, while we let \(l^-(P)\) be the perimeter of \(P\) measured clockwise. For \(K\) a compact, convex set in \(M^2\), we define \(L^{\pm}(K)=\sup l^{\pm}(P)\) after all \(P\)'s convex polygons inscribed in \(K\). Various questions regarding sharp bounds on the perimeter of \(B\) itself were posed during the years. Some have been answered. As an example it is known since the 30's (due to Golab) that, if \(B\) is symmetric with respect to \(N\), thus \(L(B)=L^+(B)=L^-(B)\), then \(6 \leq L(B) \leq 8\). There are still several open questions. In particular, Diskant asked about the lower bound of the perimeter of unit balls which are convex \(4\)- and \(5\)-gons. This has been motivated by the result \(\inf L^{\pm}(P_n)=6\), where \(P_n\) is a convex \(n\)-gon, which prompted Makeev to ask whether \(L^{\pm}(P_4)\geq 8\). In view of this, the authors of this paper prove that \[ L^{\pm}(P_4)\geq 3 \cdot (x_0+1) + \frac{1}{x_0} \approx 7.7290, \] where \(x_0\) denotes a real root of the equation \(x^3-x-1=0\).
scientific article; zbMATH DE number 5888458