Fano varieties with small non-klt locus

Let X be a Fano variety of index k such that the non-klt locus Nklt(X) is not empty. We prove that Nklt(X) has dimension at least k-1 and equality holds if and only if Nklt(X) is a linear projective space P^{k-1}. In this case X has lc singularities and is a generalised cone with Nklt(X) as vertex. If X has lc singularities and Nklt(X) has dimension k we describe the non-klt locus and the global geometry of X. Moreover, we construct examples to show that all the classification results are effective.