  We introduce a method to lower bound an entropy-based measure of genuine
multipartite entanglement via nonlinear entanglement witnesses. We show that
some of these bounds are tight and explicitly work out their connection to a
framework of nonlinear witnesses that were published recently. Furthermore we
provide a detailed analysis of these lower bounds in the context of other
possible bounds and measures. In exemplary cases we show that only a few local
measurements are necessary to determine these lower bounds.
