A monotone path connected Chebyshev set is a sun (Q2435938)

For a nonempty subset \(M\) of a normed space \(X\) denote by \(d(x,M)\) the distance from \(x\) to \(M\) and by \(P_Mx=\{y\in M :\|x-y\|=d(x,M)\}\) the (set-valued) operator of metric projection on \(M\). The set \(M\) is called proximinal (Chebyshev) if \(P_Mx\neq \emptyset\) (resp. \(P_Mx\) is a singleton) for every \(x\in X\setminus M.\) The set \(M\) is called a sun if for every \(x\in X\setminus M\) there exists \(y\in P_Mx\) such that (1)\; \(y\in P_M(y+t(y-x))\) for all \(t\geq 0\), and a strict sun if it is proximinal and (1) holds for every \(y\in P_Mx\). The set \(M\) is called an \(\alpha\)-sun if for every \(x\in X\setminus M\) there exists a ray \(\ell\) issuing form \(x\) such that \(d(z,M)=d(x,M)+\|z-x\|\) for every \(z\in\ell.\) For a survey on the problem of the convexity of Chebyshev sets and suns in normed spaces, see the paper by \textit{V. S. Balaganskij} and \textit{L. P. Vlasov} [Russ. Math. Surv. 51, No. 6, 1127--1190 (1996); translation from Usp. Mat. Nauk 51, No. 6, 125--188 (1996; Zbl 0931.41017); errata ibid. 52, No. 1, 237 (1997)] Every Chebyshev set in a finite-dimensional normed space is a path connected sun, a result that is not true in infinite dimensions. A curve \(k(\tau)\), \(0\leq\tau\leq 1,\) in a normed space \(X\) is called monotone if \(f\circ k\) is a monotone function for every \(f\in\) ext\(\, S^*\) -- the set of the extreme points of the unit sphere of the dual space \(X^*.\) The set \(M\) is called monotone path connected if any two points in \(M\) can be joined by a monotone path lying in \(M\). The main results obtained by the author in this paper are the following ones: \(\bullet\) \; a monotone path connected set in a normed space is an \(\alpha\)-sun (Theorem 1); \(\bullet\)\; a monotone path connected Chebyshev set in a normed space is a sun (Theorem 2). These theorems extend previous results of the author concerning similar properties in the spaces \(C(Q),\, Q\) a metrizable compact space, \(\ell^\infty(n)\) and \(c_0\).