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The $L_p$ cosine transform on Grassmann manifolds naturally induces finite dimensional Banach norms whose unit balls are origin-symmetric convex bodies in $\rn$. Reverse isoperimetric type volume inequalities for these bodies are established, which extend results from the sphere to Grassmann manifolds.

In this paper, we establish the $L_p$ Loomis-Whitney inequality for even isotropic measures in terms of the support function of $L_p$ projection bodies with complete equality conditions. This generalizes Ball's Loomis-Whitney inequality to the $L_p$ setting. In addition, the sharp upper bound of the minimal $p$-mean width of $L_p$ zonoids is obtained.

We establish a dual version of the Loomis-Whitney inequality for isotropic measures with complete equality conditions, where the sharp lower bound is given in terms of the volumes of hyperplane sections. For the special case of cross measures, we can drop the condition that the underlying body has centroid at the origin, yielding an independent proof of a result of Meyer's.

Corresponding to the Legendre ellipsoid and the LYZ ellipsoid, two new sine ellipsoids are introduced in this paper. These four ellipsoids are closely related in the Pythagorean relation and duality. Several volume inequalities and the valuation properties are obtained for two new ellipsoids.

The dual Loomis-Whitney inequality provides the sharp lower bound for the volume of a convex body in terms of its $(n-1)$-dimensional coordinate sections. In this paper, some reverse forms of the dual Loomis-Whitney inequality are obtained. In particular, we show that the best universal DLW-constant for origin-symmetric planar convex bodies is $1$.

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