Removing vertices from $k$-connected graphs without losing $k$-connectivity

The problem of removing vertices from a $k$-connected graph without losing $k$-connectivity is studied. We prove that one can remove some inner vertices from $k$-blocks, provided the interior of each block is large enough with respect to its boundary and the degree of any vertex of the graph is greater than $\frac{3k-1}{2}$ or $\frac{3k}{2}$.