cone
A SPHERE has no vertices and no FLAT surfaces
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.
An infinite flat surface, or an infinite surface with zero curvature.
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'.
Hemisphere
A sphere.
cone
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.
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.
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'.
A plane is a flat, closed figure.a flat surface on which a straight line joining any two points on it would wholly lie:
A flat surface that is infinitely large and with zero thickness