LVMH, the largest luxury conglomerate in the world, has been an investor in the funds of the American private equity firm Catterton for almost two decades. And when LVMH’s private equity arm, L Capital, which invests in brands in Europe and Asia, considered expanding into the United States, its American counterpart was contemplating the opposite move. Rather than compete, the two have now decided to combine, in a deal rarely seen among private equity firms.