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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 R" 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 Wj (M) ≤ Wj(K), for any 0 < j < n - i, with equality if and only if K = M. The case that K is a centered body implies Funk's section theorem.

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