In this note we exhibit some examples of proper cones that have the property of being of constant opening angle. In particular, we analyze the class of Reuleaux cones in \(\mathbb {R}^n\) with \(n\ge 3\) . Such cones are constructed as intersection of n revolutions cones \(\textrm{Rev}(g_1,\psi ),\ldots , \textrm{Rev}(g_n,\psi )\) whose incenters \(g_1,\ldots , g_n\) are unit vectors forming a common angle. The half-aperture angle \(\psi \) of each revolution cone corresponds to the common angle between the incenters. A major result of this work is that a Reuleaux cone in \(\mathbb {R}^n\) is of constant opening angle if and only if \(n= 3\) . Reuleaux cones in dimension higher than 3 are not of constant opening angle, but such mathematical objects are still of interest. In the same way that a Reuleaux triangle is a “rounded” version of an equilateral triangle, a Reuleaux cone can be viewed as a rounded version of an equiangular simplicial cone and, therefore, it has a lot of symmetry in it.
