Sets of Constant Relative Width and Constant Relative Brightness

Chakerian, G. D.

doi:10.2307/1994362

Cited by 11 publications

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“…The definition and some elementary properties of mixed projection bodies can be found in the classic volume of Bonnesen-Fenchel [6]. Support functions of mixed projection bodies were investigated by Chakerian [9]. Stability questions for mixed projection bodies are treated by Goodey [11] and Goodey-Groemer [12].…”

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Inequalities for mixed projection bodies

Lutwak¹

1993

Trans. Amer. Math. Soc.

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Abstract. Mixed projection bodies are related to ordinary projection bodies (zonoids) in the same way that mixed volumes are related to ordinary volume. Analogs of the classical inequalities from the Brunn-Minkowski Theory (such as the Minkowski, Brunn-Minkowski, and Aleksandrov-Fenchel inequalities) are developed for projection and mixed projection bodies.Two decades ago, Bolker [5] observed that projection bodies (also known as zonoids) were objects of independent investigation in a number of mathematical disciplines such as measure theory, crystallography, optimal control theory, functional analysis, and geometric convexity. Since the appearance of Bolker's article, projection bodies have received considerable increased attention (see, for example, [7,13,14,21,23,26,27,28,32,40, 41] Mixed projection bodies are related to ordinary projection bodies in the same way that mixed volumes are related to ordinary volume. The definition and some elementary properties of mixed projection bodies can be found in the classic volume of . Support functions of mixed projection bodies were investigated by Chakerian [9]. Stability questions for mixed projection bodies are treated by Goodey [11] and . In [18] and [19], inequalities for the polars of mixed projection bodies were obtained. This article treats the corresponding inequalities for the mixed projection bodies themselves. Analogs of the classical mixed volume inequalities (such as the Brunn-Minkowski, Minkowski, and Aleksandrov-Fenchel inequalities) will be established for mixed projection bodies.Since interest in zonoids is not limited to one discipline, an attempt is made to make this article reasonably self-contained.Background material and notation regarding mixed volumes and mixed surface area measures is given in §0. The classical mixed volume inequalities are

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Inequalities for mixed projection bodies

Lutwak¹

1993

Trans. Amer. Math. Soc.

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“…In dimension three, this problem has become known as Nakajima's problem [11]; see [1], [2], [3], [4], [5], [6]. It is easy to check that the answer to it is in the affirmative if K is a convex body in R 3 of class C 2 .…”

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Nakajima's Problem: Convex Bodies of Constant Width and Constant Brightness

Howard

Hug

2007

Mathematika

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ABSTRACT. For a convex body K ⊂ R n , the kth projection function of K assigns to any k-dimensional linear subspace of R n the k-volume of the orthogonal projection of K to that subspace. Let K and K 0 be convex bodies in R n , and let K 0 be centrally symmetric and satisfy a weak regularity and curvature condition (which includes all K 0 with ∂K 0 of class C 2 with positive radii of curvature). Assume that K and K 0 have proportional 1st projection functions (i.e., width functions) and proportional kth projection functions. For 2 ≤ k < (n + 1)/2 and for k = 3, n = 5 we show that K and K 0 are homothetic. In the special case where K 0 is a Euclidean ball, we thus obtain characterizations of Euclidean balls as convex bodies of constant width and constant k-brightness.

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“…For K symmetric with respect to the origin, Petty [28] has shown that the requirement that K be a dilation of n*if is equivalent to the requirement that if have constant relative brightness with respect to if (relative brightness in the sense of [28] rather than in the sense of [10]). …”

Section: Polar Centroid Inequalitiesmentioning

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“…Introductions to mixed area measures can also be found in Schneider [37], Burago and Zalgaller [7], Busemann [9] and Leichtweiss [19]. Chakerian [10] is a good reference for the mixed brightness functions considered in §2. In §3, some inequalities are derived for polars of mixed projection bodies, analogous to the Aleksandrov-Fenchel and Minkowski inequalities.…”

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Mixed projection inequalities

Lutwak¹

1985

Trans. Amer. Math. Soc.

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ABSTRACT.A number of sharp geometric inequalities for polars of mixed projection bodies (zonoids) are obtained.Among the inequalities derived is a polar projection inequality that has the projection inequality of Petty as a special case. Other special cases of this polar projection inequality are inequalities (between the volume of a convex body and that of the polar of its ¿th projection body) that are strengthened forms of the classical inequalities between the volume of a convex body and its projection measures (Quermassintegrale). The relation between the Busemann-Petty centroid inequality and the Petty projection inequality is shown to be similar to the relation that exists between the Blaschke-Santaló inequality and the affine isoperimetric inequality of affine differential geometry.Some mixed integral inequalities are derived similar in spirit to inequalities obtained by Chakerian and others. Introduction.Projection bodies or zonoids (we will use the terms interchangeably) in Rn have received considerable attention in recent years (see, for example, [6,16,25,34,36,[45][46][47][48][49]). They arise naturally in a number of disciplines such as functional analysis, convexity, the geometry of polytopes and stochastic geometry. Zonoids can be defined in several equivalent ways [4,42], for example, as the range of a nonatomic vector (R"-valued) measure or as the limit (with respect to the Hausdorff metric) of finite sums of segments. The polars of zonoids can be viewed as the finite-dimensional central sections of the unit ball of L1. The work of Bolker [4] and Schneider and Weil [42] provide excellent introductions to the subject of zonoids or projection bodies. (We follow the standard usage in convexity of the term projection body [42]. Thus what we refer to as a projection body is the polar of the object Bolker [4] calls a projection body.)The purpose of this article is to present some sharp inequalities involving mixed projection bodies. Mixed projection bodies are related to projection bodies in the same way that mixed volumes are related to ordinary volumes. The mixed projection bodies are by definition more general than projection bodies. However, the class of mixed projection bodies is the same as the class of projection bodies. The definition and elementary properties of mixed projection bodies can be found in the classic volume of Bonnesen and Fenchel [5]. The support functions of mixed projection bodies were studied by Chakerian [10] (see also [43]). Schneider [36] obtained an interesting characterization of the sphere involving mixed projection bodies.One of our objectives is to obtain a general form of the projection inequality of Petty [28] involving the volume of a convex body and that of the polar of its associ-

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Sets of Constant Relative Width and Constant Relative Brightness

Cited by 11 publications

References 4 publications

Inequalities for mixed projection bodies

Inequalities for mixed projection bodies

Nakajima's Problem: Convex Bodies of Constant Width and Constant Brightness

Mixed projection inequalities

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