Let \(\mathcal{E}\) be a Euclidean plane with Euclidean norm \(\| \dots \| \). A bounded convex centrally symmetric subset \(K\) of \(\mathcal{E}\) with a non-empty interior defines a norm \(\| \dots \| _K\) by \(\| v \| _K = \left(\inf \{r : r v \in K\}\right)^{-1}\). In fact, every norm on \(\mathcal{E}\) is of the form \(\| \dots \| _K\) for such a set \(K\). The plane endowed with a norm \(\| \dots \| _K\) is referred to as the Minkowski plane associated with \(K\), and it is called Euclidean if the norm is induced by an inner product. Its unit circle is the boundary \(C\) of \(K\). The \(K\)-length of a line segment \([x,y]\) is \(\| x - y \| _K\). A \(2n\)-gon is called \(K\)-normal if its sides have equal \(K\)-lengths and if every two opposite sides are parallel. If \(K\) is the ordinary unit circle, then \(K\)-normality is equivalent to regularity. A side-median of a polygon inscribed in a circle is the line segment joining the midpoint of one of its sides to the center of the circle. The paper under review establishes the existence of non-Euclidean Minkowski planes in which every \(K\)-normal \(2n\)-gon that is inscribed in the unit circle \(C\) has its side-medians of \(K\)-length \(\cos (2\pi/n)\), thus showing that this property does not characterize Euclidean spaces. As for the existence of \(K\)-normal \(2n\)-gons in a Minkowski space, the author elaborates on an earlier result that he had proved in [\textit{J.~Gao}, J.~Dyn. Syst. Geom. Theor. 1, 113--120 (2003; Zbl 1053.52004)]. Thus the existence and uniqueness of a \(K\)-normal \(2n\)-gon inscribed in \(C\) (and having a given point on \(C\) as one of its vertices) is guaranteed under any of the following two conditions: (i) \(n \geq 2\) and \(C\) does not contain a line segment, (ii) \(n \geq 4\) and \(C\) does not contain a line segment of \(K\)-length greater than 1. For an interesting survey of the geometry of Minkowski spaces, see [\textit{H.~Martini}, \textit{K.~J.~Swanepoel} and \textit{G.~Weiß}, Expo. Math. 19, 97--142 (2001; Zbl 0984.52004)] and [\textit{H.~Martini} and \textit{K.~J.~Swanepoel}, Expo. Math. 22, 93--144 (2004; Zbl 1080.52005)].