Criteria for unique metric lines in Banach spaces

Abstract:

A metric space $M$ has the monotone property if for each point $p$ and line $L$ of $M$ the distance $px$ between $p$ and a point $x$ of $L$ is monotone increasing as $x$ recedes along either half-line of $L$ determined by the foot of $p$ on $L$. It is shown that a Banach space (over the reals) has the monotone property if and only if it has unique metric lines. Using previously known results, additional equivalents of the monotone property are obtained and new proofs of some older criteria for unique metric lines result.