Subdifferentials of distance function outside of target set in Riemannian manifolds (Q2119293)

In this paper, some subdifferentials of distance function from a closed subset of a Riemannian manifold in points outside the target set are being studied. The exposition consists of four sections, described as below. Section 1 is introductory. Section 2 contains some preliminary material from nonsmooth analysis on Riemannian manifolds. Section 3 is devoted to computing the subdifferentials of distance function from a closed subset of a Riemannian manifold outside some target set. The following are the main result obtained there. \textbf{Theorem 1.} Let \(M\) be a Riemannian manifold and \(S\) be a nonempty closed subset of \(M\). Further, let \(x\in M\setminus S\) and put \(r:=d_S(x)\). Then, \(\partial_Fd_S(x)=N_{S(r)}^F(x)\cap \{\zeta\in T_x M; ||\zeta||=1\}\). \textbf{Theorem 2.} Let \(M\) be a Riemannian manifold and \(S\) be a nonempty closed subset of \(M\). Further, let \(x\in M\) be such that \(d_S(x)=r> 0\). Then, \(\partial_Pd_S(x)= N_{S(r)}^P(x)\cap \{\zeta\in T_x M; ||\zeta||_x=1\}\). \textbf{Theorem 3.} Let \(M\) be a Riemannian manifold and \(S\) be a nonempty closed subset of \(M\). Further, let \(x\in M\) be such that \(d_S(x)=r> 0\) and \(d_S\) is directionally regular at \(x\). Then, \(\partial_Cd_S(x)= N_{S(r)}^C(x)\cap \{\zeta\in T_x M; ||\zeta||_x=1\}\). \textbf{Theorem 4.} Let \(M\) be a Riemannian manifold and \(S\) be a nonempty closed subset of \(M\). Further, let \(x\in M\) be such that \(d_S(x)=r> 0\). Then, the following assertions are holding: (A1) If \(d_S\) is Fréchet subdifferentially regular at \(x\), then \(S(r)\) is Fréchet normally regular at \(x\) and \(d_S\) is directional regular at \(x\) (A2) If \(S(r)\) is Fréchet normally regular at \(x\) and \(d_S\) is directionally regular at \(x\), then \(d_S\) is Fréchet subdifferentially regular at \(x\). \textbf{Theorem 5.} Let \(M\) be a Riemannian manifold and \(S\) be a nonempty closed subset of \(M\). Further, let \(x\in M\) be such that \(d_S(x)=r> 0\) Then, the following assertions are holding: (A1) If \(d_S\) is proximal subdifferentially regular at \(x\), then \(S(r)\) is proximal normally regular at \(x\) and \(d_S\) is directional regular at \(x\) (A2) If \(S(r)\) is proximal normally regular at \(x\) and \(d_S\) is directionally regular at \(x\),then \(d_S\) is proximal subdifferentially regular at \(x\). Finally, Section 4 contains concluding discussions upon the obtained results. Further aspects occasioned by these developments are also being discussed.