### Well-posedness and porosity in best approximation problems

DOI: http://dx.doi.org/10.12775/TMNA.2001.041

#### Abstract

Given a nonempty closed subset $A$ of a Banach space $X$ and a point $x \in X$,

we consider the problem of finding a nearest point to $x$ in $A$.

We define an appropriate complete metric space $\mathcal M$ of all pairs

$(A,x)$ and construct a subset $\Omega$ of $\mathcal M$ which is the countable

intersection of open everywhere dense sets such that for each

pair in $\Omega$ this problem is well-posed. As a matter of fact, we

show that the complement of $\Omega$ is not only of the first category,

but also sigma-porous.

we consider the problem of finding a nearest point to $x$ in $A$.

We define an appropriate complete metric space $\mathcal M$ of all pairs

$(A,x)$ and construct a subset $\Omega$ of $\mathcal M$ which is the countable

intersection of open everywhere dense sets such that for each

pair in $\Omega$ this problem is well-posed. As a matter of fact, we

show that the complement of $\Omega$ is not only of the first category,

but also sigma-porous.

#### Keywords

Banach space; complete metric space; generic property; hyperbolic space; nearest point; porous set

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