Geodesic
On transversals of the family of translates of two-dimensional convex compact
Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 3, Tome 252 (1998), pp. 67-77

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The following theorem gives an affirmative answer to Grünbaum's old equistion. Let $\mathscr K$ be the family of translates of a convex compact set $K\subset\mathbb R^2$. If every two elements of $\mathscr K$ have a common point, then there exist three points $A,B,C\in\mathbb R^2$ such that every element of $\mathscr K$ contains some of these points. 
@article{ZNSL_1998_252_a7,
     author = {R. N. Karasev},
     title = {On transversals of the family of translates of two-dimensional convex compact},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {67--77},
     publisher = {mathdoc},
     volume = {252},
     year = {1998},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1998_252_a7/}
}
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R. N. Karasev. On transversals of the family of translates of two-dimensional convex compact. Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 3, Tome 252 (1998), pp. 67-77. http://geodesic.mathdoc.fr/item/ZNSL_1998_252_a7/