none!all shapes are flat
cone
A SPHERE has no vertices and no FLAT surfaces
The shape you are describing is a triangular prism. It has six vertices, nine edges, and five flat faces (two triangular faces and three rectangular faces). Additionally, it has no curved surfaces, as all its faces are flat.
Cool question. I think there are at least three: hemisphere, section of an obloid, cone. Since you did not say face, you appreciate difference between definitions in platonic solids as defined by Euclid and Euler, and the curved solids. A hemisphere has one flat surface and no vertices, but so does a cone. A vertex is defined as the meeting of edges, which are defined as straight in euclidean geometry. Since there are no straight edges coming to a point in a cone (unless you want to talk about infinite edges emanating from the flat surface), there are no vertices on a cone.
The shape that has zero vertices and edges and one face is a sphere. A sphere is a perfectly round three-dimensional object where every point on its surface is equidistant from its center. Unlike polygons or polyhedra, it does not have corners or edges, making it unique in its geometry.
Hemisphere
cone
A sphere.
A SPHERE has no vertices and no FLAT surfaces
cylinder
There are zero diagonals in a circle because a circle has no sides no vertices or it doesn't have a flat surface so zero.
The shape you are describing is a triangular prism. It has six vertices, nine edges, and five flat faces (two triangular faces and three rectangular faces). Additionally, it has no curved surfaces, as all its faces are flat.
Cool question. I think there are at least three: hemisphere, section of an obloid, cone. Since you did not say face, you appreciate difference between definitions in platonic solids as defined by Euclid and Euler, and the curved solids. A hemisphere has one flat surface and no vertices, but so does a cone. A vertex is defined as the meeting of edges, which are defined as straight in euclidean geometry. Since there are no straight edges coming to a point in a cone (unless you want to talk about infinite edges emanating from the flat surface), there are no vertices on a cone.
The electric field strength just outside the flat surface of a conductor is zero.
An infinite flat surface, or an infinite surface with zero curvature.
Zero vertices
Your question is inconsistent on its face. You asked "What shape has zero faces . . . . . and has only one face." I'd say that nothing could ever meet both of those requirements. They are ... how you say ... 'mutually exclusive'.