Inequalities for mixed p-affine surface area

Werner, Elisabeth M.; Ye, Deping

doi:10.1007/s00208-009-0453-2

Cited by 63 publications

(76 citation statements)

References 42 publications

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“…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%

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%

Lp Geominimal Surface Areas and their Inequalities

Ye¹

2014

International Mathematics Research Notices

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“…We only mention the rapid progress in the L p Brunn Minkowski theory (e.g., [2,4,8,10,21,22,26,27]) and the theory of valuations e.g., [5,6,7,19]. The resulting body of work has proved to be a valuable tool in fields such as harmonic analysis, information theory, stochastic geometry and PDEs (e.g., [9,14,15,25]).…”

Section: Introductionmentioning

confidence: 99%

Dual Affine invariant points

Werner

Meyer

Schütt

2015

Indiana Univ. Math. J.

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“…The renewed interest in affine invariants has benefited also from a systematic approach classifying them, as for example in [8], [14], [15], [17], and from their use in affine and affine Sobolev inequalities [10], [11], [18], [22] - [25], [27], [37], [38] and problems arising in differential geometry [4], [5], [6], [9], [21], [33] - [36] which rely on isoperimetric-type functional inequalities. It is the subject of such inequalities that is our primary goal of an on-going project.…”

Section: Introductionmentioning

confidence: 99%

Some Affine Invariants Revisited

Stancu

2013

Asymptotic Geometric Analysis

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We present several sharp inequalities for the SL(n) invariant Ω 2,n (K) introduced in our earlier work on centro-affine invariants for smooth convex bodies containing the origin. A connection arose with the Paouris-Werner invariant Ω K defined for convex bodies K whose centroid is at the origin. We offer two alternative definitions for Ω K when K ∈ C 2 + . The technique employed prompts us to conjecture that any SL(n) invariant of convex bodies with continuous and positive centro-affine curvature function can be obtained as a limit of normalized p-affine surface areas of the convex body.

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Inequalities for mixed p-affine surface area

Cited by 63 publications

References 42 publications

Lp Geominimal Surface Areas and their Inequalities

Lp Geominimal Surface Areas and their Inequalities

Dual Affine invariant points

Some Affine Invariants Revisited

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