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We consider quantitative versions of Helly-type questions, that is, insteadof finding a point in the intersection, we bound the volume of theintersection. Our first main geometric result is a quantitative version of theFractional Helly Theorem of Katchalski and Liu, the second one is aquantitative version of the $(p,q)$-Theorem of Alon and Kleitman.

Quantitative Fractional Helly and $(p,q)$-Theorems