A cone.
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One edge (boundary) and no vertices.
one
A cone.
A cone has only one edge simply because it only has one face. Although one face may have several edges, a circle is a closed figure with only one vertice, resulting in one edge. The rest of the shape is 3-D, but with only one face, there is an infinite amount of points on a cone. The cone joins on another vertice, but since the face is a circle, there is still only one actual edge. Thus, there is only one edge on a cone because there is a single, circular face.
A cone would fit the given description