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A concave polyhedron can rest on its edges.
Added 3 minutes later
Yes, it can. It will be a concave polyhedron that will roll from faces to edges.
Added 17 minutes later
Let's take, for example, a polyhedron formed by gluing two regular tetrahedra together along one face. Its center of gravity will be in the middle. This point will be the midpoint of the faces by which the tetrahedra are glued. Let's cut off almost at the very edge of the common face after gluing the tetrahedra together at both ends with a small pyramid - a regular tetrahedron. And let's hollow out three faces symmetrically inward from each side. The position of the center of gravity of the polyhedron formed in this way will not change, and if it is placed on any face after this, then the projection of the center of mass onto the plane of this face will be outside it - which means the polyhedron will roll onto its edges.
Added 3 minutes later
Yes, it can. It will be a concave polyhedron that will roll from faces to edges.
Added 17 minutes later
Let's take, for example, a polyhedron formed by gluing two regular tetrahedra together along one face. Its center of gravity will be in the middle. This point will be the midpoint of the faces by which the tetrahedra are glued. Let's cut off almost at the very edge of the common face after gluing the tetrahedra together at both ends with a small pyramid - a regular tetrahedron. And let's hollow out three faces symmetrically inward from each side. The position of the center of gravity of the polyhedron formed in this way will not change, and if it is placed on any face after this, then the projection of the center of mass onto the plane of this face will be outside it - which means the polyhedron will roll onto its edges.
Всё не так плохо как Вы думаете. Всё намного хуже!