OAI.TriangularCovering.main

The theorem states that three claims hold together in three-dimensional Euclidean space. Let H=√2 and let K be the set of points (x,y,Ht) with 0≤t≤1, |x|≤1−t and |y|≤t. For a unit vector u, shadowArea(u) is the two-dimensional Hausdorff area of the orthogonal projection of K onto the plane perpendicular to u, and A_min is the infimum of shadowArea(u) over all unit vectors u. A triangular cylinder consists of a unit axis and a compact, convex, measurable base of finite two-dimensional area that is the convex hull of three affinely independent points and is orthogonal to the axis; its carrier is the set of all points b+r·axis with b in the base and r real. First, A_min equals H. Second, the asymptotic statement MainAsymptotic holds: there exist δ>0 and C>0 such that for every ε with 0<ε<δ there is a family of 2n(ε) triangular cylinders, where n(ε)=⌈2/ε²⌉, whose carriers cover K, whose total base area divided by H differs from 1/2−(13/6000)ε² by at most Cε⁴, and whose total base area is strictly less than A_min/2. Third, NormalizedCounterexample holds: there exist a finite family of triangular cylinders covering K whose relative cost, the sum over the cylinders of base area divided by shadowArea of the cylinder's axis, is strictly less than 1/2.