Affine integral geometry and convex bodies

Abstract

SUMMARYGiven a convex body Q in Euclidean n‐dimensional space, the affine invariant measure of the set of pairs of parallel hyperplanes containing Q is an affine invariant J(Q) of Q, which, for ellipsoids, parallelepipeds and possibly for simplices of any dimensions, is proportional to V−1, where V represents the volume of Q. Consequently, for the kind of bodies mentioned, it is possible to estimate V−1 from its breadth measured in uniform random directions. If the boundary of Q is of class C2, we obtain a set of affine invariants Jh(Q) (h = any integer) which is easily calculable for ellipsoids. In particular, J‐n(Q) coincides with J(Q) and J‐(n+1)(Q) is the affine invariant measure of all pairs of parallel hyperplanes that ‘support’ Q as described by Firey(1972, 1985), Schneider (1978, 1979) and Weil (1979, 1981). For a general convex body Q the values of Jh(Q) cannot be expressed in terms of simple metric invariants (such as volume, surface area, breadth, width) and this justifies the study in the last section of certain inequalities between them.