The Randomized Integer Convex Hull

Let K ⊂ R d be a sufficiently round convex body (the ratio of the circumscribed ball to the inscribed ball is bounded by a constant) of a sufficiently large volume. We investigate the randomized integer convex hull I L (K) = conv(K ∩L), where L is a randomly translated and rotated copy of the integer lattice Z d. We estimate the expected number of vertices of I L (K), whose behaviour is similar to the expected number of vertices of the convex hull of VolK random points in K. In the planar case we also describe the expectation of the missed area Vol(K \I L (K)). Surprisingly, for K a polygon, the behaviour in this case is different from the convex hull of random points.