# On Non-separable Families of Positive Homothetic Convex Bodies

## Abstract

A finite family \(\mathcal {B}\) of balls with respect to an arbitrary norm in \(\mathbb R^d\) (\(d\ge 2\)) is called a non-separable family if there is no hyperplane disjoint from \(\bigcup \mathcal {B}\) that strictly separates some elements of \(\mathcal {B}\) from all the other elements of \(\mathcal {B}\) in \(\mathbb R^d\). In this paper we prove that if \(\mathcal {B}\) is a non-separable family of balls of radii \(r_1, r_2,\ldots , r_n\) (\(n\ge 2\)) with respect to an arbitrary norm in \(\mathbb R^d\) (\(d\ge 2\)), then \(\bigcup \mathcal {B}\) can be covered by a ball of radius \(\sum _{i=1}^n r_i\). This was conjectured by Erdős for the Euclidean norm and was proved for that case by Goodman and Goodman (Am Math Mon 52:494–498, 1945). On the other hand, in the same paper Goodman and Goodman conjectured that their theorem extends to arbitrary non-separable finite families of positive homothetic convex bodies in \(\mathbb R^d\), \(d\ge 2\). Besides giving a counterexample to their conjecture, we prove that conjecture under various additional conditions.

## Keywords

Convex body Positive homothets Non-separable family*k*-Impassable family

## Mathematics Subject Classification

52C17 05B40 11H31 52C45## Notes

### Acknowledgments

Károly Bezdek: Partially supported by a Natural Sciences and Engineering Research Council of Canada Discovery Grant. Zsolt Lángi: Partially supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences and the OTKA K_16 Grant 119495. The authors would like to thank the anonymous referees for careful reading and valuable comments and proposing a shortcut in the proof of Theorem 4.

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