Geometric applications of Chernoff-type estimates and a ZigZag approximation for balls

Abstract

In this paper we show that the euclidean ball of radius 1 1 in R n \mathbb {R}^n can be approximated up to ε > 0 \varepsilon >0 , in the Hausdorff distance, by a set defined by N = C ( ε ) n N = C(\varepsilon )n linear inequalities. We call this set a ZigZag set, and it is defined to be all points in space satisfying 50 50% or more of the inequalities. The constant we get is C ( ε ) = C ln ⁡ ( 1 / ε ) / ε 2 C(\varepsilon ) = C \ln (1/\varepsilon )/\varepsilon ^2 , where C C is some universal constant. This should be compared with the result of Barron and Cheang (2000), who obtained N = C n 2 / ε 2 N = Cn^2/\varepsilon ^2 . The main ingredient in our proof is the use of Chernoff’s inequality in a geometric context. After proving the theorem, we describe several other results which can be obtained using similar methods.