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Explore Petty's and Brannen's conjectures, Zhang's and Petty's inequalities, and Minkowski's insights on mixed volumes in geometric tomography, with a focus on ellipsoids and simplices. Delve into Lp-centroid bodies and an extension of Lutwak's inequality. Proceeding from Petty's and considering functional minimization with homothetic ellipsoids for level sets, discover insights on Petty's projection inequality. Learn about Lutwak's inequality extension and more.
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(d-1)-dimensional Hausdorff measure where σK is the area measure of K where x is the segment joining x to the origin where n(x) is the outward unit normal at x (from Geometric Tomography, R.J.Gardner, 1995)
What about the volume of ΠK ? Petty’s conjecture: ellipsoids are minimizers Brannen’s conjecture: simplices are maximizers
* Petty’s inequality: ellipsoids are minimizers Zhang’s inequality: simplices are maximizers
A proof of Petty’s inequality. Ingredients: Mixed volumes. Minkowski’s first inequality: V1(K, L)d V(K)d-1V(L) , and equality holds if and only if K and L are homothetic. Lp-centroid body. Lp-Busemann-Petty centroid inequality: V(ΓpK) V(K) , and equality holds if and only if K is a centered ellipsoid. Lutwak, Yang, Zhang (2000), Campi, G. (2002)
A proof of Petty’s inequality. Consider the functional We have with equality if and only if K and L are homothetic ellipsoids, with L origin-symmetric.
A proof of Petty’s inequality. minimized when L is a level set of hΠK , so when L is a dilatation of Π*K
Petty’s projection inequality Petty’s conjecture Lutwak’s inequality