Central Points for Sets in Rn (or: the Chocolate Ice-Cream Problem)

Let A be a subset of the unit ball in R n , and 0 r 1 real. Find a point x for which the intersection of the r-neighborhood of x with A has a large measure. Tight bounds on this measure are found. 1 General After dinner a round bowl of chocolate-and-vanilla ice-cream is served. percent of the ice-cream is chocolate, your favorite avor. You have a round scoop with which to probe the dessert and your goal is to maximize the percentage of chocolate ice-cream in your scoop. How well can you do? More formally, let A be a measurable subset of the unit ball B in R n. We seek, for every r 1 an r-central point x r-i.e. a point whose r-neighborhood has a large intersection with A. Speciically we aim to maximize (A\Br(xr)) (A). This question can also be turned around: What is the largest measure of A B in R n that intersects every ball of radius r in a set of measure t ? As stated, this is a natural question in integral geometry. For analogous problems in the realm of nite graphs, see LPRS]. We also consider the existence of a central point x for which (A\Br(x)) (A) is reasonably large for all r 1. For both questions, we seek answers that hold for every measurable set A. Our results are presented in the following two theorems: