# 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)|

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

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