On Well Posed Best Approximation Problems for a Nonsymmetric Seminorm

\def\inter{\mathop{\rm int}} Let $M$ be a closed convex (generally unbounded) subset of a Banach space $E$ with $0$ being an interior point of $M$, $A$ be a closed subset of $E$. Let $T_{M}(A)$ be the set of all $x_{0}\in E$ such that the problem $\smash{\min\limits_{a\in A}}\, \mu_{M} (x_{0}-a)$ is well posed, where $\mu_{M}$ is the Minkowski functional of $M$, so $\mu_{M}$ is a nonsymmetric seminorm. We obtain some asymptotic properties (appearance far from the origin) of $M$ which are necessary and/or sufficient for $S_{M}^{\inter}(A)\setminus T_{M}(A)$ to be a meagre or a $\sigma$-porous subset of $$S_{M}^{\inter}(A)=\left\{x_{0}\in E\Big|\ 0<\varrho_{M}(x_{0},A)<\sup\limits_{x\in E}\varrho_{M}(x,A)\right\}\,$$ where $\varrho_{M}(x,A)=\inf\limits_{a\in A}\mu_{M}(x-a)$.