A three-dimensional solid is a convex set if it contains every line segment connecting two of its points. A convex polyhedron is a polyhedron that, as a solid, forms a convex set.

Why are Platonic solids called that?

Geometers have studied the Platonic solids for thousands of years. They are named for the ancient Greek philosopher Plato who hypothesized in one of his dialogues, the Timaeus, that the classical elements were made of these regular solids.

Are all polyhedrons convex?

All regular polyhedra (i.e., Platonic solids) are convex.