Euclidean cones over circles (Q2706957)

In Aleksandrov's theory of metric spaces a Euclidean cone \(C(X)\) over the metric space \((X,d_X)\) is constructed defining a metric \(d_{C(X)}\) on \(C(X):= X\times[0, \infty)/(x,0) \sim(y,0)\) by NEWLINE\[NEWLINEd^2_{C(X)} ((x,s), (y,t)): =s^2+t^2 -2st\cos (\min\{d_X(x, y),\pi\})NEWLINE\]NEWLINE [\textit{A. D. Aleksandrov}, \textit{V. Berestovskii} and \textit{I. Nikolaev}, Russ. Math. Surveys 41 (3), 1-54 (1986; Zbl 0625.53059) p. 17 with \(K=0]\). In the special case of \(X\) being the circle \(S^1(r)\) of radius \(r>0\) in the Euclidean plane with usual metric the authors investigate some properties of the unit disc \(B_r(\overline x,1)\) of \(C(S^1(r))\) with centre \(\overline x=(x,t) (t\neq 0)\). They concern the Euclidean convexity and area of \(B_r(\overline x,1)\).