On the limited power of linear probes and other optimization oracles

Throughout this note, X denotes a real vector space of finite dimension d _> 2. As the term is used here, a convex body in X is a compact convex subset C of X whose interior is nonempty. When the origin 0 is interior to C, the gauge functional of C is the positively homogenous function gc whose value is 1 at all points of C's boundary bd C. This function is subadditive, and is symmetric (gc(x) = gc(-x)) precisely when C itself is symmetric about 0 (i.e. C = -C). In this case, gc is a norm for the space X. Here we consider the problem of using norm-computations to find a convex polytope P (not necessarily in any sense the smallest one) that contains C. By finding P we mean to produce its vertex-set or, equivalently for our model, the set of hyperplanes determined by P's facets. It is assumed that the usual arithmetic operations in R and the usual vector operations in X are available without cost, so the problem's difficulty is measured solely in terms of the number of calls to the "oracle" that computes norms. Computing the norm of a point x E X \ {0} is equivalent to finding the point at which the ray [0, co)x intersects the boundary bd C. Thus we refer to C's ray-oracle (Pc, which, when presented with any ray R issuing from 0, returns the point R N (bd C). Such probing oracles are currently of widespread interest because of their use in robotics and tomography (see, e.g.,)