Characterizations of some rotundity properties in terms of farthest points
Arunachala Prasath C, Vamsinadh Thota
TL;DR
The paper develops a program to characterize classical rotundity notions in Banach spaces through sets of farthest and almost farthest points and the generalized diameter $r(A,B)$. By introducing and relating remotality notions (remotal, uniquely remotal, SUR, USUR, sup-compact) to the geometry of the unit sphere and ball via $Q_F(x)$, $Q_F(x,\delta)$, and $P_A(x,\delta)$, it derives precise equivalences: $X$ is rotund iff $S_X$ is uniquely remotal on $X\setminus\{0\}$; $X$ is CLUR iff $S_X$ is sup-compact on $X\setminus\{0\}$ (with corresponding convergence of $Q_{S_X}(x,\tfrac{1}{n})$); UR and LUR admit uniform characterizations using $X_\alpha$ and $USUR$ on $X_\alpha$. The results unify rotundity concepts with proximinality-type notions and provide generalized-diameter criteria and Chebyshev-type descriptions, offering geometric tools for best-approximation theory in Banach spaces.
Abstract
We characterize rotund, uniformly rotund, locally uniformly rotund and compactly locally uniformly rotund spaces in terms of sets of (almost) farthest points from the unit sphere using the generalized diameter. For this we introduce few remotality properties using the sets of almost farthest points. As a consequence, we obtain some characterizations of the aforementioned rotundity properties in terms of existing proximinality notions.