Solution of the illumination problem for three-dimensional convex bodies.
Let \(a\) be a boundary point of a convex body \(M\subset {\mathbb R}^n\) and \(e\) be a non-zero vector. If for sufficiently small \(\lambda >0\) the point \(a+\lambda e\) belongs to the interior of \(M\), then we say \(a\) is illuminated by the direction of the vector \(e\). The directions of non-zero vectors \(e_{1},e_{2},\dots ,e_{p}\) illuminate the boundary of the body \(M\) if each of its boundary points is illuminated by at least one of these directions. The least integer \(p\) of the non-zero vectors \(e_{1},e_{2},\dots ,e_{p}\) whose directions illuminate the boundary of \(M\) is denoted by \(c(M)\). The functional \(md\) is defined as follows: Let \(M\subset {\mathbb R}^n\) be a compact, convex body; the greatest integer \(m\) for which there exist \(m+1\) regular boundary points of \(M\) such that the outward unit normals \(w_{0},\dots ,w_{m}\) of \(M\) at these points are minimally dependent is denoted by \(mdM\). The author first formulates his main conjecture: If \(M\subset {\mathbb R}^n\) is a compact convex body with \(mdM=m\geq 2\) then \(c(M)\leq 2^{n}-2^{n-m}\) and then proceeds to prove that this conjecture holds for \(mdM=2\) (Theorem I) and for \(mdM=n\) (Theorem II).
scientific article; zbMATH DE number 2074708