Cross $i$-sections of star bodies and dual mixed volumes

In this paper, we establish an extension of Funk's section theorem. Our result has the following corollary: If $K$ is a star body in $\mathbb{R}^n$ whose central $i$-slices have the same volume (with appropriate dimension) as the central $i$-slices of a centered body $M$, then the dual quermassintegrals satisfy $\widetilde{W}_j(M)\leq \widetilde{W}_j(K)$, for any $0\leq j<n-i$, with equality if and only if $K=M$. The case that $K$ is a centered body implies Funk's section theorem.