$L_p$ John Ellipsoids

Lutwak, Erwin; Yang, Deane; Zhang, Gaoyong

doi:10.1112/s0024611504014996

Cited by 165 publications

(118 citation statements)

References 34 publications

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“…Note that our definition of −p K is different from the definition given by Lutwak, Yang and Zhang in [11]. That is for K = B, we have −p B = B.…”

mentioning

confidence: 58%

“…will be called the dual L p -John ellipsoid of K, denoted byẼ p K. Lutwak, Yang, and Zhang [11] showed that when p ≥ 1, the volume of E p K is always dominated by the volume of K.…”

mentioning

confidence: 99%

“…Lutwak, Yang, and Zhang (see [11]) have generalized the classical John ellipsoid JK to the L p -John ellipsoid E p K which can be associated with a fixed convex body K: If K ∈K n o and p > 0, amongst all origin-centered ellipsoids E, the unique ellipsoid E p K that solves the constrained maximization problem …”

mentioning

confidence: 99%

“…If K ∈K n o and p ≥ 1, then the L p -polar projection body −p K is an originsymmetric body whose radial function is given by [11] …”

mentioning

confidence: 99%

“…For many notions, there are natural extensions of projection bodies, centroid bodies, John ellipsoids etc. in the L p -Brunn-Minkowski theory (see [7,9,11]). …”

mentioning

confidence: 99%

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VOLUME INEQUALITIES FOR L_p-JOHN ELLIPSOIDS AND THEIR DUALS

Lu¹,

Leng²

2007

Glasgow Math. J.

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Abstract. In this paper, we establish some inequalities among the L p -centroid body, the L p -polar projection body, the L p -John ellipsoid and its dual, which are the strengthened version of known results. We also prove inequalities among the polar of the L p -centroid body, the L p -polar projection body, the L p -John ellipsoid and its dual.2000 Mathematics Subject Classification. 52A40, 52A20. In this paper, we continue the study of the volume inequalities among the L pcentroid body the L p -polar projection body, the L p -John ellipsoid and its dual.The setting for this paper is n-dimensional Euclidean space ‫ޒ‬ n . Let K n denote the set of convex bodies (compact, convex subsets with non-empty interiors). Let K

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“…Note that our definition of −p K is different from the definition given by Lutwak, Yang and Zhang in [11]. That is for K = B, we have −p B = B.…”

mentioning

confidence: 58%

“…will be called the dual L p -John ellipsoid of K, denoted byẼ p K. Lutwak, Yang, and Zhang [11] showed that when p ≥ 1, the volume of E p K is always dominated by the volume of K.…”

mentioning

confidence: 99%

mentioning

confidence: 99%

“…If K ∈K n o and p ≥ 1, then the L p -polar projection body −p K is an originsymmetric body whose radial function is given by [11] …”

mentioning

confidence: 99%

“…For many notions, there are natural extensions of projection bodies, centroid bodies, John ellipsoids etc. in the L p -Brunn-Minkowski theory (see [7,9,11]). …”

mentioning

confidence: 99%

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VOLUME INEQUALITIES FOR L_p-JOHN ELLIPSOIDS AND THEIR DUALS

Lu¹,

Leng²

2007

Glasgow Math. J.

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Lp$L_p$‐cosine transform of the (p,q)${(p,q)}$‐th dual curvature measure

Fang

Wang

Zhang

2022

Mathematische Nachrichten

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The (𝑝, 𝑞)-th dual curvature measure was introduced by Lutwak, Yang and Zhang. In this paper, we study the 𝐿 𝑝 cosine transform of the (𝑝, 𝑞)-th dual curvature measure which defines a new convex body. We prove that the new convex body unifies the 𝐿 𝑝 Petty projection body and the 𝐿 𝑝 centroid body. The corresponding affine isoperimetric inequality is also obtained. It is an extension of the known 𝐿 𝑝 Petty projection inequality. K E Y W O R D S𝐿 𝑝 centroid body, 𝐿 𝑝 cosine transform, 𝐿 𝑝 Petty projection body, (𝑝, 𝑞)-th dual curvature measure M S C ( 2 0 2 0 ) 52A20, 52A40

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