Being Fat and Friendly is Not Enough

We show that there is no $(1+\text{eps})$-approximation algorithm for the problem of covering points in the plane by minimum number of fat triangles of similar size (with the minimum angle of the triangles being close to 45 degrees). Here, the available triangles are prespecified in advance. Since a constant factor approximation algorithm is known for this problem \cite{cv-iaags-07}, this settles the approximability of this problem. We also investigate some related problems, including cover by friendly fat shapes, and independent set of triangles in three dimensions.