The Steiner formula for Minkowski valuations

Parapatits, Lukas; Schuster, Franz E.

doi:10.1016/j.aim.2012.03.024

Cited by 73 publications

(46 citation statements)

References 65 publications

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“…The aim of this paper is to obtain the Brunn-Minkowski-type inequalities with respect to Blaschke L p harmonic combinations for quermassintegrals and dual quermassintegrals of L p moment bodies (see e.g. [1,2,26,31,32] for related results).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

BRUNN–MINKOWSKI TYPE INEQUALITIES FOR L_p MOMENT BODIES

Zhu

Zhou

2013

Glasgow Math. J.

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About 15 years ago, Lutwak and Zhang (E. Lutwak and G. Zhang, Blaschke-Santalo inequalities, J. Differ. Geom. 47 (1997), 1-16) introduced the notion of L p moment bodies and established important volume inequalities for them, which were recently generalized by Haberl and Schuster (C. Haberl and E. Schuster, General L p affine isoperimetric inequalities, J. Differ. Geom. 83 (2009), 1-26). In this paper, we establish new Brunn-Minkowski-type inequalities with respect to Blaschke L p harmonic addition for the quermassintegrals and dual quermassintegrals of L p moment bodies.2000 Mathematics Subject Classification. 52A40. Introduction and main results.Centroid bodies were defined and investigated by Petty [27]. They have proven to be a remarkably powerful tool in establishing a number of fundamental affine isoperimetric inequalities due to Petty [27][28][29] (see also [18,19]). Projection bodies were introduced by Minkowski at the turn of the previous century and have since become a central notion in convex geometry (see e.g.[8] and the references therein). All centroid and projection bodies are zonoids. However, the centroid operator and the projection operator are quite different. For example, the centroid operator commutes with linear transformations, while the projection operator does not; the projection operator is translation invariant, but the centroid operator is not.While projection bodies and the projection operator have been the objects of intensive investigations during more than 50 years, centroid bodies (volumenormalized moment bodies) and their L p extensions by Lutwak and Zhang [24] have attracted increased attention only in the last two decades (see e.g. [4-6, 8-15, 19-25, 33, 35, 36, 38]). The aim of this paper is to obtain the Brunn-Minkowski-type inequalities with respect to Blaschke L p harmonic combinations for quermassintegrals and dual quermassintegrals of L p moment bodies (see e.g. [1, 2, 26, 31, 32] for related results).Let K n denote the set of convex bodies (compact, convex subsets with non-empty interiors) in ‫ޒ‬ n . For the set of convex bodies containing the origin in their interior and the set of convex bodies centred at the origin, we write K n o and K n s respectively. The unit ball in ‫ޒ‬ n and its surface will be denoted by B and S n−1 respectively. The volume and surface area of the convex body K will be denoted by V (K) and S(K) respectively.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

BRUNN–MINKOWSKI TYPE INEQUALITIES FOR L_p MOMENT BODIES

Zhu

Zhou

2013

Glasgow Math. J.

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“…Recently, Parapatits and the first author [50] proved that for any Φ ∈ MVal, there exist Φ (j) ∈ MVal, where 0 ≤ j ≤ n, such that…”

Section: Smooth and Generalized Valuationsmentioning

confidence: 99%

Minkowski valuations and generalized valuations

Schuster

Wannerer

2018

J. Eur. Math. Soc.

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A convolution representation of continuous translation invariant and SO(n) equivariant Minkowski valuations is established. This is based on a new classification of translation invariant generalized spherical valuations. As applications, Crofton and kinematic formulas for Minkowski valuations are obtained., the subspace of SO(n − 1) invariant valuations in Val i . In turn, valuations in Val SO(n−1) i are spherical.

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“…, n − 1}, where W (n−1) i (K|u ⊥ ) denotes the ith quermassintegral of K|u ⊥ defined in the subspace u ⊥ . For more information about mixed projection bodies, we refer to Gardner [11, p. 185], Lutwak [21,23], Parapatits and Schuster [33], and Schneider [36, p. 578].…”

Section: 4mentioning

confidence: 99%

A unified treatment for $L_p$ Brunn-Minkowski type inequalities

Zou

Xiong

2018

Communications in Analysis and Geometry

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A unified approach used to generalize classical Brunn-Minkowski type inequalities to L p Brunn-Minkowski type inequalities, called the L p transference principle, is refined in this paper. As illustrations of the effectiveness and practicability of this method, several new L p Brunn-Minkowski type inequalities concerning the mixed volume, moment of inertia, quermassintegral, projection body and capacity are established.2010 Mathematics Subject Classification: 52A40.

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The Steiner formula for Minkowski valuations

Cited by 73 publications

References 65 publications

BRUNN–MINKOWSKI TYPE INEQUALITIES FOR L_p MOMENT BODIES

BRUNN–MINKOWSKI TYPE INEQUALITIES FOR L_p MOMENT BODIES

Minkowski valuations and generalized valuations

A unified treatment for $L_p$ Brunn-Minkowski type inequalities

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The Steiner formula for Minkowski valuations

Cited by 73 publications

References 65 publications

BRUNN–MINKOWSKI TYPE INEQUALITIES FOR Lp MOMENT BODIES

BRUNN–MINKOWSKI TYPE INEQUALITIES FOR Lp MOMENT BODIES

Minkowski valuations and generalized valuations

A unified treatment for $L_p$ Brunn-Minkowski type inequalities

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Product

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BRUNN–MINKOWSKI TYPE INEQUALITIES FOR L_p MOMENT BODIES

BRUNN–MINKOWSKI TYPE INEQUALITIES FOR L_p MOMENT BODIES