An asymmetric seminorm is a positive, positively homogeneous and subadditive functional \(\|\cdot|\) on a real linear space \(X\), or on a cone \(X\) (also called a semilinear space). Asymmetric semimetrics \(\rho\) (or quasi-metrics) are considered as well (see, for instance, [the reviewer, Functional analysis in asymmetric normed spaces. Basel: Birkhäuser (2013; Zbl 1266.46001)].
The author extends Michael’s selection theorem, in both symmetric and asymmetric cases, to set-valued mappings taking values in the class of \(\overset{\circ}{B}\)-infinitely connected subsets, a class that properly contains that of convex subsets. Applications are given to the set-valued operators of best and of near-best approximation.
He also considers \(\varepsilon\)-selections \(\varphi:X\to M\) of the metric projection on a convex subset \(M\) of an asymmetric normed space \((X,\|\cdot|)\), meaning additive \(\varepsilon\)-selections, satisfying \(\rho(x,\varphi(x)\le\rho(x,M)+\varepsilon\), or multiplicative ones, such that \(\rho(x,\varphi(x)\le(1+\varepsilon)\rho(x,M),\) where \(\rho(x,M)=\inf\{\|y-x|:y\in M\}.\)
Another result proved in the paper is that in the space of all convex non-empty bounded subsets of a reflexive normed space \(X\) equipped with the Hausdorff semimetric, every bounded set admits a Chebyshev center.