Convex Hypersurfaces with Hyperplanar Intersections of Their Homothetic Copies

摘要

Extending a well-known characteristic property of ellipsoids, we describe all convex solids K subset of R-n, possibly unbounded, with the following property: for any vector z is an element of R-n and any scalar lambda not equal 0 such that K not equal z + lambda K, the intersection of the boundaries of K and z + lambda K lies in a hyperplane. This property is related to hyperplanarity of shadow-boundaries of K and central symmetricity of small 2-dimensional sections of K.