The phase length of a smooth curve and the Fenchel-Reshetnyak inequality (Q1963374)

Let \(L\) be a \(C^2\)-regular closed curve in a Euclidean space. The well-known Fenchel integral inequality states that (1) \(\int_L k(s) ds\geq 2\pi\), where \(k(s)=|x''(s)|\) is the curvature of the curve \(L\) given by an equation \(x=x(s)\) with arc length parameter \(s\). If \(L\) is a curve with endpoints \(A\) and \(B\), then the following inequality holds for the curvature integral: (2) \(\int_L k(s) ds\geq \alpha+\beta\), where \(\alpha\) and \(\beta\) are the angles between the chord \(AB\) and the tangents to the curve \(L\) at the endpoints \(A\) and \(B\), respectively. This result was proven earlier by \textit{A.~D.~Aleksandrov} and \textit{Yu.~G.~Reshetnyak} [Sib. Math. J. 29, 1-16 (1988; Zbl 0639.53002)]. In the article under review, new proofs are given for the inequalities (1) and (2). The proofs are based on the concepts of the phase length of a curve and the phase distance between vectors.