The cross-section body, plane sections of convex bodies and approximation of convex bodies, I

Abstract

For a convex body K⊂ℝ d we investigate three associated bodies, its intersection body IK (for 0∈int K), cross-section body CK, and projection body IIK, which satisfy IK⊂CK⊂IIK. Conversely we prove CK⊂const1(d)I(K−x) for some x∈int K, and IIK⊂const2 (d)CK, for certain constants, the first constant being sharp. We estimate the maximal k-volume of sections of 1/2(K+(-K)) with k-planes parallel to a fixed k-plane by the analogous quantity for K; our inequality is, if only k is fixed, sharp. For L⊂ℝ d a convex body, we take n random segments in L, and consider their ‘Minkowski average’ D. We prove that, for V(L) fixed, the supremum of V(D) (with also n∈N arbitrary) is minimal for L an ellipsoid. This result implies the Petty projection inequality about max V((IIM)*), for M⊂ℝ d a convex body, with V(M) fixed. We compare the volumes of projections of convex bodies and the volumes of the projections of their sections, and, dually, the volumes of sections of convex bodies and the volumes of sections of their circumscribed cylinders. For fixed n, the pth moments of V(D) (1≤p<∞) also are minimized, for V(L) fixed, by the ellipsoids. For k=2, the supremum (n∈N arbitrary) and the pth moment (n fixed) of V(D) are maximized for example by triangles, and, for L centrally symmetric, for example by parallelograms. Last we discuss some examples for cross-section bodies.