Tomography of Convex and Star Bodies

Abstract

New definitions of a star body and its radial function are given. These are used in studying sections of convex and star bodies, in conjunction with the dual mixed volumes of Lutwak, the spherical Radon transform, and the i-chord functions. (When the body contains the origin in its interior, and i ≠ 0, the latter equals the sum of the values of the ith power of the radial function at antipodal points in the unit sphere.) Special cases of results obtained include the following. A simple construction is given of non-spherical convex bodies of constant i-section, i.e., such that each i-dimensional section through an interior point has the same i-dimensional volume. It is shown that two star bodies, containing the origin in their interiors and with analytic boundaries, must be equal up to reflection in the origin, if the volumes of their sections by i-dimensional subspaces for two different values of i are equal. Examples are given to show that the last statement is no longer necessarily true if the boundaries are only infinitely differentiable or if the bodies are convex polytopes. It is also proved that a star body which has constant i-section for two different values of i, with respect to the same interior point, must be a ball centered at that point. This answers the dual of the old question as to whether there exist non-spherical convex bodies of constant width and constant brightness. New results are also obtained concerning the determination of convex polytopes by the volumes of their sections through one or more points.