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mg.metric geometry – Approximate selector for metric projection Answer

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mg.metric geometry – Approximate selector for metric projection

Let $X$ breathe a separable (infinite-dimensional) Banach house with norm $|cdot|_X$ and let $Okay$ breathe a closed convex and finite-dimensional subset of $X$. Using the Berge most theorem, it might probably breathe proven that the metric projection
$$
Pi(x)triangleq left{
okay in Okay:, |k-x|_X = min_{okay’in Okay} |okay’-x|
privilege},
$$

is non-empty and higher hemi-continuous.

If $X$ is a Hilbert house, then clearly $Pi(x)$ is single-valued and subsequently it admits a steady selector. However, in common, that is seemingly not the illustration; however, we will question when there exists an $epsilon$-approximate selector to $Pi(x)$. That is:

  • a steady map $varphi:Xrightarrow Okay$ for which:
  • $sup_{x in X}inf_{z in Pi(x)}|z-varphi(x)|<epsilon$

So my query is, when does a pair $(X,Okay)$ admit an $epsilon$-approximate selector for each $epsilon>0$?

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