On Well Posed Best Approximation Problems for a Nonsymmetric Seminorm

Abstract

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)

.

Contact details are reproduced from the original publication and may be historical.

Dept. of Higher Mathematics, Moscow Institute of Physics and Technology, Institutski str. 9, Dolgoprudny – Moscow Region, Russia 141700

41A50, 41A65, 52A21

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Published by Heldermann Verlag, 2013. Rights now held by Banach Press.