Non-optimality of conical parts for Newton's problem of minimal resistance in the class of convex bodies

We consider Newton's problem of minimal resistance. Our main result shows that certain conical parts contained in the boundary of a convex body inhibit the optimality of the body. This is achieved by the investigation of a local variation of the conical part. We also consider the problem arising in the limit if the height goes to infinity. We apply the main result to certain bodies which are conjectured to being optimal in the literature and we show that they cannot be optimal.