Abstract
I can indicate the type of refinement mentioned in the title by referring to Kirchberger's theorem [4]. Its picturesque form in the plane is: if sheep and goats are grazing in a field and for every four animals there exists a line separating the sheep from the goats then there exists such a line for all the animals. The refinement is that the words ‘every four animals’ may be replaced by ‘every four animals including an arbitraily chosen animal’; this reduces the ‘Kirchberger number’ from four to, effectively, three.
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