Estimating the Sizes of Convex Bodies from Projections

Let 1 ⩽ r ⩽ s ⩽ d −1, and let L s = ( L 1 , α 1 ;…; L n , α n ) be a finite family of s ‐dimensional linear subspaces L i of E d , with associated positive weights α i , for i = 1,…, n . Denote by V r ( K , L ) the intrinsic r ‐volume of the image of the at least r ‐dimensional compact convex set K under orthogonal projection on to L , write and let V r ( K ) be the intrinsic r ‐volume of K . (The intrinsic r ‐volumes, which are normalized quermassintegrals, are measures of the sizes of convex sets in various senses.) In this paper is considered the problem of determining bounds for the ratio π r ( K , L s ) = V r ( K , L s )/ V r ( K ), or, in other words, the stereometric problem of how good an estimator V r ( K , L s ) is for V r ( K ). In the case when r = s = 1 and L 1 has a large amount of symmetry, a complete solution is given. In the case when r = s = d − 1; there is also a solution which is, in principle, complete. More generally, estimates can be made when s = d − 1; this problem is related to another on approximating the unit ball by zonotopes.