Given a point in $m$-dimensional objective space, any $\varepsilon$-ball of a point can be partitioned into the incomparable, the dominated and dominating region. The ratio between the size of the incomparable region, and the dominated (and dominating) region decreases proportionally to $1/2^{m-1}$, i.e., the volume of the Pareto dominating orthant as compared to all other volumes. Due to this reason, it gets increasingly unlikely that dominating points can be found by random, isotropic mutations. As a remedy to stagnation of search in many objective optimization, in this paper, we suggest to enhance the Pareto dominance order by involving an obtuse convex dominance cone in the convergence phase of an evolutionary optimization algorithm. We propose edge-rotated cones as generalizations of Pareto dominance cones for which the opening angle can be controlled by a single parameter only. The approach is integrated in several state-of-the-art multi-objective evolutionary algorithms (MOEAs) and tested on benchmark problems with four, five, six and eight objectives. Computational experiments demonstrate the ability of these edge-rotated cones to improve the performance of MOEAs on many-objective optimization problems.