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

Being Fat and Friendly is Not Enough