Can a convex polyhedron with $3$ vertices, $2$ faces and $3$ edges exist? If so, how does it look like or what's its name? My imagination just fails here...
I was reading a proof of Euler's formula (https://plus.maths.org/content/eulers-polyhedron-formula) where it boils down to showing that $V+F-E$ for the polyhedron in question equals $2$. But I'm not really sure if such a convex polyhedron can exist at all.
Depends on what you mean by "polyhedron". If you, like me, want the faces and edges to be flat and straight, then this is impossible. You need at least four faces to make a polyhedron.
If you allow faces and edges to be curved, then you can imagine gluing two equilateral triangles edge to edge, and then "inflate" it a bit. Or, equivalently, take a sphere, call the equator an edge, and put three vertices along the same equator.