Angle sum of polygons in space (Q6172862)

Consider a closed polygonal line \(L\) obtained as the union of \(n\) line segments glued at their vertices and contained in the \(d\)-dimensional Euclidean space where \(d\geq3\). Any of these segments forms an angle of at most \(\pi\) with the next segment (which can be measured in a plane that contains them both). The authors remark in this note that, by the spherical triangle inequality, the sum \(\alpha(L)\) of these angles (over the \(n\) segments \(L\) is made up of) satisfies \[ 0\leq\alpha(L)\leq(n-2)\pi \text{ when } n \text{ is even } \] and \[ \pi\leq\alpha(L)\leq(n-2)\pi \text{ when } n \text{ is odd. } \] Inversely, for any angle \(\gamma\) in these intervals (\([0,(n-2)\pi]\) when \(n\) is even and \([\pi,(n-2)\pi]\) when \(n\) is odd), a polygonal line \(L\) formed from \(n\) line segments and contained in the \(3\)-dimensional Euclidean space is constructed explicitly, that satisfies \(\gamma=\alpha(L)\). It is also shown that, when \(\gamma\) is at the boundary of these intervals, any line \(L\) such that \(\gamma=\alpha(L)\) is contained in a plane, and that, when \(\gamma\) is equal to \((n-2)\pi\), that line must be the boundary of a convex polygon.