Let \(q\) be any point and \(M\) a compact convex body of \(\mathbb {R}^{n}\). The image of \(M\) under the homothety with center \(q\) and ratio \(k\), where \(k\) is a real number such that \(0<k<1\), is called the diminished copy of \(M\). The least integer \(p\) such that \(M\) can be covered by \(p\) diminished copies of \(M \) is denoted by \(b(M)\). Gohberg and Markus observing that \(b(M)=2^{n}\) for every \(n\)-dimensional parallelotope stated that \(b(M)\leq 2^{n}\) for every compact, convex body \(M\subset \mathbb {R}^{n}\) and equality holds only for parallelotopes.
V. Boltyanskii [Izv. Mold. Fil. Akad. Nauk SSSR 10(76), 79-86 (1960)] gave an equivalent formulation of this problem, known as the Gohberg-Markus-Hadwiger problem: A boundary point \(a\) of a convex body \(M\) of \(\mathbb {R}^{n}\) is illuminated by the direction of a nonzero vector \(e\) of \(\mathbb {R}^{n}\) if for \(\lambda >0\) small enough the point \(a+\lambda e\) belongs to the interior of \(M\). Furthermore the directions of non-zero vectors \( x\in bdM\) is illuminated by at least one of these directions. The least integer \(p\) such that there exist nonzero vectors \(e_{1},\dots ,e_{p}\) whose directions illuminate the boundary of \(M\) is denoted by \(c(M)\).
The theorem, formerly proven, stating that for every compact convex body \(M\subset \mathbb {R}^{n}\), the equality \(b(M)=c(M)\) holds, helps to give another form of the Gohberg-Markus-Hadwiger problem: Prove \(c(M)\leq 2^{n}\) for every compact convex body \(M\subset \mathbb {R}^{n}\), equality holding only for parallelotopes.