Hypersphere

A hypersphere is a highly-symmetric object in each respective dimension. Hyperspheres are not polytopes, as they are round objects with no vertices or other flat elements. The surface of the hypersphere in n dimensions is the set of all points a fixed distance away from a given point (its center)

It is the most common practice to refer to a hypersphere by the dimensionality of its surface, which is one less than the space it occupies. For example, a 3-sphere refers to the hypersphere in 4D space (a glome).

Examples
Excluding the point, the hyperspheres up to 10D are the following: