The maximum relative diameter for multi-rotationally symmetric planar convex bodies (Q2794825)

A plane convex body \(C\) is called \(k\)-rotationally symmetric if it is invariant under the rotation of angle \(\frac{2\pi}{k}\) around a certain point, and it is called multi-rotationally symmetric if it is \(k\)-rotationally symmetric for more than one value of \(k\). Given a decomposition of a \(k\)-rotationally symmetric plane convex body \(C\) into \(k\) connected subsets \(C_1,C_2,\dots, C_k\), the maximum relative diameter of \(C\) is defined as NEWLINE\[NEWLINE d_M(P) = \max \{ D(C_i) : i=1,2,\dots, k \} , NEWLINE\]NEWLINE where \(D(C_i)\) is the diameter of \(C_i\). A decomposition is called a \(k\)-partition if it is defined by \(k\) simple curves, starting at an interior point of \(C\) and ending at a boundary point of \(C\). A \(k\)-partition is standard if these curves are rotated copies of a segment starting at the centre of rotation of \(C\). It is proved in the paper [the author, \textit{U. Schnell} and \textit{S. Segura Gomis}, ``Subdivisions of rotationally symmetric planar convex bodies minimizing the maximum relative diameter'', Preprint, \url{arXiv:1501.03907}] that over the family of \(k\)-partitions, \(d_M(P)\) is minimal for a standard \(k\)-partition of \(C\).NEWLINENEWLINEThe author in this paper investigates multi-rotationally symmetric plane convex bodies. Let \(C\) be \(k_i\)-rotationally symmetric for \(k_1 < k_2 < \dots < k_n\). Let \(d_M(P_{k_i})\) denote the minimum of the maximum relative diameter of \(C\) over the family of \(k_i\)-partitions, where \(P_{k_i}\) denotes the standard \(k_i\)-partition for which this minimum is attained. Then, clearly, NEWLINE\[NEWLINE d_M(P_{k_1}) \geq d_M(P_{k_2}) \geq \dots \geq d_M(P_{k_n}). NEWLINE\]NEWLINENEWLINENEWLINEThe main results of the author are as follows. {\parindent=6mm \begin{itemize} \item[(1)] If \(k_1 \geq 3\), then the above inequalities are a chain of equalities if, and only if \(k_1 \geq 7\). \item [(2)] If \(k_1 \geq 3\), then the above inequalities are a chain of equalities if, and only if \(k_n\) is a product of (not necessarily distinct) prime numbers, each greater than or equal to \(7\). \item [(3)] If \(k_1 = 2\), then \(d_M(P_{k_1}) > d_M(P_{k_2}) \geq \dots \geq d_M(P_{k_n})\).NEWLINENEWLINE\end{itemize}}