2022/01/09 by Silouanos Brazitikos, Brazitikos, Silouanos, Dimitris-Marios Liakopoulos +1
Biochemistry, Genetics and Molecular Biology · Mathematics · #Diffusion and Search Dynamics #FOS: Mathematics #Geometric Analysis and Curvature Flows #Metric Geometry (math.MG) #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2201.03093
openalex publication_date 2022/01/09 · openalex created_date 2022/04/03 · openalex updated_date 2026/07/28
We study the slicing inequality for the surface area instead of volume. This is the question whether there exists a constant αn depending (or not) on the dimension n so that S(K)≤αn|K|(1)/(n)maxξ∈ Sn-1S(K∩ξ⊥ ) where S denotes surface area and |⋅ | denotes volume. For any fixed dimension we provide a negative answer to this question, as well as to a weaker version in which sections are replaced by projections onto hyperplanes. We also study the same problem for sections and projections of lower dimension and for all the quermassintegrals of a convex body. Starting from these questions, we also introduce a number of natural parameters relating volume and surface area, and provide optimal upper and lower bounds for them. Finally, we show that, in contrast to the previous negative results, a variant of the problem which arises naturally from the surface area version of the equivalence of the isomorphic Busemann--Petty problem with the slicing problem has an affirmative answer.