Ref: More on shadows of 3D convex bodies,
Shadows and planar sections of polyhedra
Given a 3D convex body C, we define a maximal area (perimeter) section of C with respect to any specified direction $n$ in space as the maximum area (perimeter) planar section with $n$ normal to the plane of section. Is the following claim valid?
- If all maximal perimeter sections of C have equal perimeter, then, C is a sphere.
(Convex bodies with constant maximal section function in odd dimensions discusses the corresponding claim with all max area cross sections of C having same area).
Note: One can also possibly think of convex bodies such that all maximal area sections have same perimeter and vice versa.
