Covering a set with $N$ balls of common minimal(!) radius (Existence of such a covering)

I don't feel too bad posting a "proof by references" since I worked on this problem for quite a while but failed, before realizing the $N=1$ case seems pretty google-able. In any event, it seems Konyagin proved that any non-reflexive Banach space admits an equivalent norm and some 3 points so that there is no smallest ball with respect to the equivalent norm containing those 3 points. A good buzzword is "Chebyshev center". Below is a link to a strengthening of Konyagin's result (and a mention of Konyagin's result). I couldn't find a free version of Konyagin's paper online.

https://dml.cz/bitstream/handle/10338.dmlcz/119230/CommentatMathUnivCarolRetro_42-2001-1_11.pdf

Tags:

Optimization

Geometry

Banach Spaces

Functional Analysis

Reflexive Space

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