We investigate numerically the optimal constants in Lieb-Thirring inequalities by studying the associated maximization problem. We use a monotonic fixed-point algorithm and a finite element discretization to obtain trial potentials which provide lower bounds on the optimal constants. We examine the one-dimensional and radial cases in detail. Our numerical results provide new lower bounds, insight into the behavior of the maximizers and confirm some existing conjectures. Based on our numerical results, we formulate a complete conjecture about the best constants for all possible values of the parameters.
