The convex floating body

Floating Body Cell

Floating Body, Illumination Body, and Polytopal Approximation

Let K be a convex body in Rd and Kt its floating bodies. There is a polytope that satisfies Kt ⊂ Pn ⊂ K and has at most n vertices, where n ≤ e vold(K \Kt) t vold(B d 2 ) . Let Kt be the illumination bodies of K and Qn a polytope that contains K and has at most n (d−1)-dimensional faces. Then vold(K t \K) ≤ cd vold(Qn \K), where n ≤ c dt vold(K t \K).

On Convex Body Chasing

A player moving in the plane is given a sequence of instructions of the following type: at step i a planar convex set Fi is specified, and the player has to move to a point in Fi. The player is charged for the distance traveled. We provide a strategy for the player which is competitive, i.e., for any sequence Fi the cost to the player is within a constant (multiplicative) factor of the "off-lin...

Inequalities for the Difference Body of a Convex Body

In the following, 5 will denote the boundary of the unit ball in En, and u a variable point of S (so re is a unit vector, or "direction"). The polar equation of the boundary of DK is given by p = /»(«), uES, so piu) is the radius of DK in the direction u. Then p(w) is the maximum length of a chord of K having direction u—the length of a "diameter" of K having direction u. Let p. denote w-dimens...

Computing Convex Hull in a Floating Point Arithmetic

We present a numerically stable and time and space complexity optimal algorithm for constructing a convex hull for a set of points on a plane. In contrast to already existing numerically stable algorithms which return only an approximate hull, our algorithm constructs a polygon that is truly convex. The algorithm is simple and easy to implement.