There is no such poyhedron or "normal" object.
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There is no simply connected polyhedron or a curved equivalent that meets these requirements.
5 edges, 5 verts, and faces are only on 3d shapes, for the most part, so any 2d shape has 1
5
A square based pyramid.
A pentagon based pyramid.
There is no such shape. The only convex shape that has four faces is a tetrahedron, and that has four vertices, not 5.