Inequalities for the Surface Area of Projections of Convex Bodies

Abstract

Abstract. We provide general inequalities that compare the surface area S(K) of a convex body K in R n to the minimal, average or maximal surface area of its hyperplane or lower dimensional projections. We discuss the same questions for all the quermassintegrals of K. We examine separately the dependence of the constants on the dimension in the case where K is in some of the classical positions or K is a projection body. Our results are in the spirit of the hyperplane problem, with sections replaced by projections and volume by surface area.

On the average volume of sections of convex bodies

“…Closing this introductory section we would like to note that the results of this article are dual to the ones in [9]. In that work, the main question was to compare the surface area S(K) of a convex body K in R n to the minimal, average or maximal surface area of its hyperplane or lower dimensional projections.…”

“…is the minimal surface area parameter of K. Another result from [9] asserts that if K is in some of the classical positions mentioned above, then…”

“…In that work, the main question was to compare the surface area S(K) of a convex body K in R n to the minimal, average or maximal surface area of its hyperplane or lower dimensional projections. One of the main results in [9] states that there exists an absolute constant c 1 > 0 such that, for every convex body K in R n ,…”

On the Reverse Loomis–Whitney Inequality

“…For instance, [9] and [16] consider versions involving projections on coordinate subspaces of any dimension, [1] establishes a connection with Shearer's entropy inequality, [2] gives a generalization for projections along other suitable directions, [7], [8], [11] prove extensions to analytical inequalities, [6] shows its relevance for the multilinear Kakeya conjecture, and [37] studies its implications in group theory. The very recent papers [12], [13], [32] give various generalizations particularly in the context of the uniform cover inequality, for the average volume of sections or between quermassintegrals of convex bodies and extremal or average projections on lower dimensional spaces, respectively.…”

Συναρτησιακές Ανισότητες Και Εφαρμογές Σε Προβλήματα Κυρτής Και Στοχαστικής Γεωμετρίας

“…για κάθε κυρτό σώμα K στον R n και κάθε u ∈ S n−1 , όπου S(A) είναι η επιφάνεια του A στην κατάλληλη διάσταση. Η ανισότητα αυτή χρησιμοποιήθηκε στο [43] για τη μελέτη ενος ερωτήματος των Dembo, Cover και Thomas [34] σχετικά με τη μονοτονία ενός ανάλογου της πληροφορίας Fisher στην κλάση των συμπαγών κυρτών συνόλων, και εμφανίζεται ξανά στο [44] όπου μελετάται το ερώτημα να συγκριθεί η επιφάνεια S(K) ενός κυρτού σώματος K στον R n με τη μέση, ελάχιστη ή μέση επιφάνεια των προβολών του συνδιάστασης 1.…”