Quantitative Helly-type theorems
Quantitative Helly-type theorems
We establish some quantitative versions of Helly’s famous theorem on convex sets in Euclidean space. We prove, for instance, that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper C"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">C</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {C}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is any finite family of convex sets in <inline-formula …