# Find a vector space and a set without Tschebyschev center.

I have such exercise: Give example of a vector space G and its subset A such that the center of A does not exists.

By the Tschebyschev center we mean the center of the smallest ball containing A ( that ball's radius is called radius of set A).

I spent 2 hours on thinking and havent got it yet. I think it must be infinite dimentional vector space, otherwise center exists. I thought about spaces known from functional analysis like L1,l1,c,c0,c00, but was unable to find an example.

Thanks in advance for help.

• I think you must first give as an hypothesis that $A$ is a bounded set – Jean Marie Jan 17 at 22:01
• There is a good book that contains many "Counterexamples in Analysis" (Gelbaum and Olmsted): pdfs.semanticscholar.org/a4e7/… Maybe not for what you are looking for, but in general, you can browse this book and learn many things... – Jean Marie Jan 17 at 22:05
• This paper may be useful to you as it gives examples of sets with no Chebyshev center. – Carl Christian Jan 18 at 8:25
• @JeanMarie I realised myself that A must be bounded. Otherwise, every point is a center. I treated it as obvious so I did not write it. – Maciej Ficek Jan 18 at 18:17
• Thanks @CarlChristian that example seems legit. – Maciej Ficek Jan 18 at 18:34