Symmetric \(k\)-ellipses with limiting eccentricity (Q1120828)

Consider a sequence of ellipses in which each new member has as foci the two points of intersection of the previous ellipse with its minor axis and which passes through the two points of intersection of that ellipse with its major axis. In 1986, \textit{J. N. Kapur} [``The Golden Ellipse'', J. Math. Education (1986)] has shown that the eccentricity of the ellipses in this sequence approaches the value \(2/(\sqrt{5}+1)\) (reciprocal of the Golden Ratio). For integer \(k\geq 2\), the locus of points \(X\in R_2\) \((R_2\) two-dimensional Euclidean space) for which the sum of the \(k\)-Euclidean distances between \(X\) and the vertices \(F_1,\ldots,F_k\) of a regular \(k\)-polygon in \(R_2\) is equal to a constant \(c\), is called a symmetric \(k\)-ellipse (having \(F_1,\ldots,F_k\) as ``foci''). Generalizing Kapur's result the authors construct for any finite \(k\geq 2\) a sequence of symmetric \(k\)-ellipses (which are ellipses iff \(k=2)\) such that eccentricity (which can be defined consistently with the concept of eccentricity of the 2-ellipses) of the \(k\)-ellipses in the sequence approaches a limit \(e_k\) for each \(k\); for these sequences \(e_k\) approaches the limit \(\pi/4\) as \(k\) approaches infinity. Remark: \(k\)-ellipses are important in connection with location decision problems.