On Well Posed Best Approximation Problems for a Nonsymmetric Seminorm

Let

be a closed convex (generally unbounded) subset of a Banach space

E E

with

0 0

being an interior point of

M M

,

A A

be a closed subset of

E E

. Let

T_{M}(A) T M ( A )

be the set of all

x_{0}\in E x 0 ∈ E

such that the problem

\smash{\min\limits_{a\in A}}\, \mu_{M} (x_{0}-a) min ⁡ a ∈ A   μ M ( x 0 − a )

is well posed, where

\mu_{M} μ M

is the Minkowski functional of

M M

, so

\mu_{M} μ M

is a nonsymmetric seminorm. We obtain some asymptotic properties (appearance far from the origin) of

M M

which are necessary and/or sufficient for

S_{M}^{\mathop{\rm int}}(A)\setminus T_{M}(A) S M i n t ( A ) ∖ T M ( A )

to be a meagre or a

\sigma σ

-porous subset of

S_{M}^{\mathop{\rm int}}(A)=\left\{x_{0}\in E\Big|\ 0<\varrho_{M}(x_{0},A)<\sup\limits_{x\in E}\varrho_{M}(x,A)\right\}\, S M i n t ( A ) = { x 0 ∈ E ∣   0 < ϱ M ( x 0 , A ) < sup ⁡ x ∈ E ϱ M ( x , A ) }  

where

\varrho_{M}(x,A)=\inf\limits_{a\in A}\mu_{M}(x-a) ϱ M ( x , A ) = inf ⁡ a ∈ A μ M ( x − a )

.