It depends on how an edge is defined. But using an everyday understanding of the word, the answer could be a section of a sphere, ellipsoid, paraboloid, hyperboloid, or a cone. There are many other possibilities.
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A cone has 1 circular edge
There is no solid that has those characteristics. To have an edge it must have another surface.
A cone
A single edge does not intersect!
A sphere is a solid with no edges.