Surface bodies and p-affine surface area

Schütt, Carsten; Werner, Elisabeth M.

doi:10.1016/j.aim.2003.07.018

Cited by 161 publications

(168 citation statements)

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“…The notion of L p affine surface area was further extended to all p ∈ R for general convex bodies in, e.g., [27,35,36]. In fact, extensions of L p affine surface area to all −n = p ∈ R were obtained by their integral expressions (see e.g.…”

Section: Introduction and Overview Of Resultsmentioning

confidence: 99%

“…In fact, extensions of L p affine surface area to all −n = p ∈ R were obtained by their integral expressions (see e.g. Theorem 3.1) and by investigating the asymptotic behavior of the volume of certain families of convex bodies [16,26,27,35,36,37,41,42] (and even star-shape bodies [43]). The L p affine surface area is now thought to be at the core of the rapidly developing L p -Brunn-Minkowski theory.…”

Section: Introduction and Overview Of Resultsmentioning

confidence: 99%

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Lp Geominimal Surface Areas and their Inequalities

Ye¹

2014

International Mathematics Research Notices

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In this paper, we introduce the L p geominimal surface area for all −n = p < 1, which extends the classical geominimal surface area (p = 1) by Petty and the L p geominimal surface area by Lutwak (p > 1). Our extension of the L p geominimal surface area is motivated by recent work on the extension of the L p affine surface area -a fundamental object in (affine) convex geometry. We prove some properties for the L p geominimal surface area and its related inequalities, such as, the affine isoperimetric inequality and a Santaló style inequality. Cyclic inequalities are established to obtain the monotonicity of the L p geominimal surface areas. Comparison between the L p geominimal surface area and the p-surface area is also provided.2010 Mathematics Subject Classification: 52A20, 53A15 Introduction and Overview of ResultsThe classical isoperimetric problem asks: what is the minimal area among all convex bodies (i.e., convex compact subsets with nonempty interior) K ⊂ R n with volume 1? The solution of this old problem is now known as the (classical) isoperimetric inequality, namely, the minimal area is attained at and only at Euclidean balls with volume 1. The isoperimetric inequality is an extremely powerful tool in geometry and related areas. Note that the classical isoperimetric inequality does not have the "affine invariant" flavor, because the area may change under linear transformations (even) with unit (absolute value of) determinant.However, many objects in (affine) convex geometry are invariant under invertible linear transformations. A typical example is the Mahler volume product M(K) = |K||K• |, the product of the volume of K and its polar body K• . That is, M(K) = M(T K) for all invertible linear transformations T on R n . One can ask a question for M(K) similar * Keywords: affine surface area, L p affine surface area, geominimal surface area, L p geominimal surface area, L p -Brunn-Minkowski theory, affine isoperimetric inequalities, the Blaschke-Santaló inequality, the Bourgain-Milman inverse Santaló inequality.

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Section: Introduction and Overview Of Resultsmentioning

confidence: 99%

Section: Introduction and Overview Of Resultsmentioning

confidence: 99%

Lp Geominimal Surface Areas and their Inequalities

Ye¹

2014

International Mathematics Research Notices

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“…Within the last few years, amazing connections have been discovered between some of these affine invariant notions and concepts from information theory, e.g., [13,15,18,29,31,32,43], leading to a totally new point of view and introducing a whole new set of tools in the area of convex geometry. In particular, it was observed in [53] that one of the most important affine invariant notions, the L p -affine surface area for convex bodies [27,50], is Rényi entropy from information theory and statistics. Rényi entropies are special cases of f -divergences.…”

Section: Introductionmentioning

confidence: 99%

Mixedf-divergence and inequalities for log-concave functions

Caglar

Werner

2014

Proceedings of the London Mathematical Society

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Mixed f -divergences, a concept from information theory and statistics, measure the difference between multiple pairs of distributions. We introduce them for log concave functions and establish some of their properties. Among them are affine invariant vector entropy inequalities, like new Alexandrov-Fenchel type inequalities and an affine isoperimetric inequality for the vector form of the Kullback Leibler divergence for log concave functions.Special cases of f -divergences are mixed L λ -affine surface areas for log concave functions. For those, we establish various affine isoperimetric inequalities as well as a vector Blaschke Santaló type inequality.

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“…During the past decade various elements of the L p Brunn-Minkowski theory have attracted increased attention (see e.g. [3], [4], [5], [8], [9] [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [22], [24], [25], [26], [27], [28], [29]). …”

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confidence: 99%

On the Lp Minkowski Problem for Polytopes

Hug

Lutwak²,

Yang³

et al. 2005

Discrete Comput Geom

186

147

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Abstract. Two new approaches are presented to establish the existence of polytopal solutions to the discrete-data L p Minkowski problem for all p > 1.As observed by Schneider [23], the Brunn-Minkowski theory springs from joining the notion of ordinary volume in Euclidean d-space, R d , with that of Minkowski combinations of convex bodies. One of the cornerstones of the Brunn-Minkowski theory is the classical Minkowski problem. For polytopes the problem asks for the necessary and sufficient conditions on a set of unit vectors u 1 , . . . , u n ∈ S d−1 and a set of real numbers α 1 , . . . , α n > 0 that guarantee the existence of a polytope, P , in R d with n facets whose outer unit normals are u 1 , . . . , u n and such that the facet whose outer unit normal is u i has area (i.e.,

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Surface bodies and p-affine surface area

Abstract: The surface body is a generalization of the floating body. Its relation to p-affine surface area is studied.

Cited by 161 publications

References 20 publications

Lp Geominimal Surface Areas and their Inequalities

Lp Geominimal Surface Areas and their Inequalities

Mixedf-divergence and inequalities for log-concave functions

On the Lp Minkowski Problem for Polytopes

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