Projection Bodies and Valuations

Let Π be the projection operator, which maps every polytope to its projection body. It is well known that Π maps the set of polytopes, P, in R into P, that it is a valuation, and that for every P ∈ P, ΠP is affinely associated to P . It is shown that these properties characterize the projection operator Π. This proves a conjecture by Lutwak. Let Kn denote the set of convex bodies (i.e., of compact, convex sets) in Euclidean n-space Rn and let Pn denote the set of convex polytopes in Rn. A convex body K ∈ Kn is determined by its support function, h(K, ·), on the unit sphere Sn−1, where h(K,u) = max{u · x : x ∈ K} and where u · x denotes the standard inner product of u and x. The projection body, ΠK, of K is the convex body whose support function is given for u ∈ Sn−1 by h(ΠK,u) = vol(K|u⊥), where vol denotes (n− 1)-dimensional volume and K|u⊥ denotes the image of the orthogonal projection of K onto the subspace orthogonal to u. Projection bodies were introduced by Minkowski at the turn of the last century in connection with Cauchy’s surface area formula. They are an important tool for studying projections and have also proved to be useful in other ways and in other subjects. One important aspect here is the range of the operator Π. Projection bodies of convex polytopes are special polytopes called zonotopes. These are important due to the connection to oriented matroids, hyperplane arrangements, aspects of optimization, computational geometry, and other areas (cf. [35], [5]). Projection bodies of convex bodies are highly symmetric centered convex bodies called zonoids. These arise in a number of guises; for example, the zonoids in Rn are precisely the ranges of non-atomic Rn-valued measures, and they are precisely the polars of the unit balls of n-dimensional subspaces of L1([0, 1]) (cf. the surveys [6], [31], [11]).