Minkowski sums and isoperimetric inequalities for polar moments of inertia of plane convex regions
نویسنده
چکیده
These notes are to be regarded as a ‘plea for help’. Although I continue to make progress at proofs, the progress is slow. I am out of my usual area of work, and others, more expert in convex geometry matters, may well be able to tell me whether things that I’m trying to prove are true or not, or unknown. My email is [email protected] Assistance with settling the questions associated with these notes would be much appreciated. Change log.
منابع مشابه
On Hadwiger’s results concerning Minkowski sums and isoperimetric inequalities for moments of inertia
Consider convex plane domains D(t) = (1− t)D0 + tD1, 0 ≤ t ≤ 1. We first prove that the 1/4-power of the polar moment of inertia about the centroid of D(t) is concave in t. From this we deduce some isoperimetric inequalities.
متن کاملStability for the Brunn-minkowski and Riesz Rearrangement Inequalities, with Applications to Gaussian Concentration and Finite Range Non-local Isoperimetry
We provide a simple, general argument to obtain improvements of concentration-type inequalities starting from improvements of their corresponding isoperimetric-type inequalities. We apply this argument to obtain robust improvements of the Brunn-Minkowski inequality (for Minkowski sums between generic sets and convex sets) and of the Gaussian concentration inequality. The former inequality is th...
متن کاملOptimum Quantization and Its Applications
Minimum sums of moments or, equivalently, distorsion of optimum quantizers play an important role in several branches of mathematics. Fejes Tóth’s inequality for sums of moments in the plane and Zador’s asymptotic formula for minimum distortion in Euclidean d-space are the first precise pertinent results in dimension d ≥ 2. In this article these results are generalized in the form of asymptotic...
متن کاملOrlicz Projection Bodies
As Schneider [50] observes, the classical Brunn-Minkowski theory had its origin at the turn of the 19th into the 20th century, when Minkowski joined a method of combining convex bodies (which became known as Minkowski addition) with that of ordinary volume. One of the core concepts that Minkowski introduced within the Brunn-Minkowski theory is that of projection body (precise definitions to fol...
متن کاملThe Dual Brunn-minkowski Theory for Bounded Borel Sets: Dual Affine Quermassintegrals and Inequalities
This paper develops a significant extension of E. Lutwak’s dual Brunn-Minkowski theory, originally applicable only to star-shaped sets, to the class of bounded Borel sets. The focus is on expressions and inequalities involving chord-power integrals, random simplex integrals, and dual affine quermassintegrals. New inequalities obtained include those of isoperimetric and Brunn-Minkowski type. A n...
متن کامل