Primitive inner illuminating systems for convex bodies (Q1573685)

Let \(K\subset E^d\) be a convex body, possibly unbounded. A set \(F\subset bd K\) is said to illuminate a convex body \(K\) from within provided for every point \(x\in bd K\) there is a point \(y\in F\) distinct from \(x\) such that the open line segment \(xy\) is contained in int \(K\). A set \(F\) illuminating a convex body \(K\) from within is called primitive if no proper subset of \(F\) illuminates \(K\) from within. Such a set \(F\) is said to primitively illuminate \(K\) from within. Grünbaum conjectured that any set primitively illuminating a compact, convex body \(K\subset E^d\), \(d\geq 3\), from within has at most \(2^d\) points. V. Soltan confirmed this conjecture for the case \(d=3\). More exactly, he proved that any set primitively illuminating a compact, convex body in \(E^3\) from within has at most 8 points, and only convex polytopes combinatorially equivalent to the 3-cube have primitive illuminating sets of 8 points (placed at their vertices). V. Boltyanski, H. Martini, and V. Soltan gave a negative answer to Grünbaum's conjecture for all \(d\geq 4\), by giving an example of a compact, convex body \(K\subset E^d\) which has a primitive illuminating set of at least \(m\) points for any positive integer \(m\). In this paper, the authors study Grünbaum's conjecture for the case of unbounded, convex bodies in \(E^d\) (distinct from a cone) and prove the following. Theorem 1. Any inner illuminating set of an unbounded, convex body \(K\subset E^2\) contains a subset of two points primitively illuminating \(K\) from within. Theorem 2. Any inner illuminating set of an unbounded, convex body \(K \subset E^3\) contains a subset of at most 6 points primitively illuminating \(K\) from within. Theorem 3. There is an unbounded, convex body \(K\subset E^d\), \(d\geq 4\), such that for any positive integer \(m\), \(K\) has a finite set of at least \(m\) points primitively illuminating \(K\) from within. Theorem 4. A compact, convex body \(K\subset E^3\) is a convex polytope combinatorially equivalent to the 3-cube if and only if it has a primitive inner illuminating set consisting of at least 7 points.