A new proof of a theorem of Petersen

Let $M$ be an $n$-dimensional complete Riemannian manifold with Ricci curvature $ge n-1$. In cite{colding1, colding2}, Tobias Colding, by developing some new techniques, proved that the following three condtions: 1) $d_{GH}(M, S^n)to 0$; 2) the volume of $M$ ${text{Vol}}(M)to{text{Vol}}(S^n)$; 3) the radius of $M$ ${text{rad}}(M)topi$ are equivalent. In cite{peter}, Peter Petersen, by developing a different technique, gave the 4-th equivalent condition, namely he proved that the $n+1$-th eigenvalue of $M$ $lambda_{n+1}(M)to n$ is also equivalent to the radius of $M$ ${text{rad}}(M)topi$, and hence the other two. In this note, we give a new proof of Petersens theorem by utilizing Coldings techniques.