Various topological constructions have their metric counterparts; for example, metric product corresponds to topological product and metric inverse limit corresponds to topological inverse limit. The main subject of this paper is a metric counterpart of the topological notion of quotient space. The author restricts the consideration to a compact metric space \(A\) and a very simple equivalence relation, with only one nontrivial equivalence class \(C\subset A.\) Then, evidently, the natural quotient map \(p:A\to A/C\) (defined by \(p(x)=[x])\) satisfies the condition: \(p(C)\) is a singleton and \(p|(A\smallsetminus C)\) is a topological embedding. The class of star bodies in \(\mathbb R^n\) with nonnegative and continuous radial maps is considered. The radial quotient maps are defined and corresponding quotient bodies are studied. In particular, it is proved that the operation \((A,C)\to A/C\) preserves the class of intersection bodies of star bodies. The author considers the category St\(^n\) of star bodies and star maps, and its subcategory St\(_+^n\) with star bodies whose radial functions are positive. A duality on this subcategory: a functor from St\(_+^n\) to itself, which turns out to be an involution, is introduced.