A convex compact body $K$ in 3-space has well-defined volume, surface area, and mean width. Do these quantities enable one to say anything about the "mean cross-sectional area"?

I put the phrase in quotes because there are at least two natural ways to define the term. We could average with respect to the measure on the affine Grassmannian (which is noncompact but that's okay since the set of planes intersecting $K$ is compact in Graff) or with respect to the linear Grassmannian (where within each family of parallel planes we average the cross-sectional area over all those planes that intersect $K$). Note that the former is a weighted version of the latter in which the weight is the width of $K$ in the direction perpendicular to the family of hyperplanes.

Is either of these two notions of mean cross-sectional area expressible in terms of volume, surface area, and mean width?