Mathematical word for geometrical object? Is there a mathematical word to designate the concept of a geometrical object like: * square * cube * tesseract * N-dimensional cube * circle * sphere * hypersphere * regular and non-regular polygons * regular and non-regular polyhedrons * ... What is the common word to designate all these objects ?

They are all compact subsets of suitable $\mathbb R^n$. I would call them manifolds in a broad sense, considering both the cases "manifold with corners"-"manifold without corners" (but with boundary).