Covering and guarding polygons using \(L_ k\)-sets (Q2640882)

Let P be a simple, singly-connected polygon with non-empty interior. Denote by \(\ell (x,y)\) the minimum number of segments in a polygonal chain in P that joins \(x\in P\) and \(y\in P\). We say that P is \(L_ k\)- convex if \(\ell (x,y)\leq k\) for every x,y\(\in P\). Polygons \(P_ 1,...,P_ t\subset P\) are called guards of P using \(L_ j\)-visibility if for every \(y\in P\) there is an \(x\in P_ 1\cup...\cup P_ t\) such that \(\ell (x,y)\leq j\). Here is a special case of a general theorem proved in the paper: there exist \(\lfloor n/(k+2j+1)\rfloor\) \(L_ k\)- convex n-gons as guards of P using \(L_ j\)-visibility. An example shows that the estimate cannot be improved.