This paper generalizes certain existing isoperimetric-type inequalities from R2 to higher dimensions. These inequalities provide lower bounds for the n-dimensional volume and, respectively, surface area of certain star-shaped bodies in Rn and characterize the equality cases.
More specifically, we work with g-chordal star-shaped bodies, a natural generalization of equichordal compact sets. A compact set in Rn is said to be equichordal if there exists a point in the interior of the set such that all chords passing through this point have equal
length. To justify the significance of our results, we provide several means of constructing
g-chordal star-shaped bodies.
The method used to prove the above inequalities is further employed in finding new lower bounds for the dual quermassintegrals of g-chordal star-shaped sets in Rn and, more generally, lower bounds for the dual mixed volumes involving these star bodies. Finally, some of the previous results will be generalized to Ln-stars, star-shaped sets whose radial functions are n-th power integrable over the unit sphere Sn−1.