Stability of Geometric Inequalities

Groemer, H.

doi:10.1016/b978-0-444-89596-7.50009-2

Cited by 46 publications

(48 citation statements)

References 30 publications

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“…Letting (70) and using the concavity of the resulting function, we obtain a Brunn-Minkowski inequality for quermassintegrals: If K and L are convex bodies in R n and 0 ≤ i ≤ n − 1, then…”

Section: A Surveymentioning

confidence: 99%

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The Brunn-Minkowski inequality

Gardner

2002

Bull. Amer. Math. Soc.

746

561

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Abstract. In 1978, Osserman [124] wrote an extensive survey on the isoperimetric inequality. The Brunn-Minkowski inequality can be proved in a page, yet quickly yields the classical isoperimetric inequality for important classes of subsets of R n , and deserves to be better known. This guide explains the relationship between the Brunn-Minkowski inequality and other inequalities in geometry and analysis, and some applications.

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Section: A Surveymentioning

confidence: 99%

“…Several strong forms of the Brunn-Minkowski inequality hold in special circumstances, for example, the stability estimates due to V. Diskant, H. Groemer, and R. Schneider referred to in [70,Section 3] and [135, p. 314], and an inequality of Ruzsa [132].…”

Section: S(k L ·) = S(k ·) + S(l ·)mentioning

confidence: 99%

The Brunn-Minkowski inequality

Gardner

2002

Bull. Amer. Math. Soc.

746

561

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“…First, the large body of stability results for geometric inequalities which can be traced back at least to Minkowski and Bonnesen. More recent systematic contributions are due to Volkov, Diskant, Fuglede, Groemer, Goodey, Schneider, and others, see Groemer's [11] survey. Second, a group of results of Groemer [9], [10], [12], [13], Goodey and Groemer [7], and Burger and Schneider [4] on the stability of geometric characterizations or properties of special convex bodies such as (solid) balls, ellipsoids, or prototiles.…”

Section: 1mentioning

confidence: 99%

Stability of Blaschke's Characterization of Ellipsoids and Radon Norms

Gruber¹

1997

Discrete Comput Geom

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Roughly speaking, in this article we prove quantitative versions of the following statements: (i) If each shadow boundary of a convex body in E d (d ≥ 3) under parallel illumination contains a curve which is almost planar and has the same shadow as the shadow boundary, then the body is approximately ellipsoidal. (ii) If for each diameter of a convex disk in E 2 there is a diameter which is not far from being conjugate to the former, then the disk is close to a Radon disk. A consequence of these theorems is a stability result for the symmetry of orthogonality in finite-dimensional normed spaces. In addition, a stability result for the relation between different definitions of length in a normed plane is given.

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“…For further results regarding the determination of convex bodies by certain data of their projections see [4] and [14]. Concerning the concept of stability see [6] and the survey [7].…”

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confidence: 99%

On the girth of convex bodies

Groemer

1997

Archiv der Mathematik

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For a three-dimensional convex body and a given direction the corresponding girth is defined as the perimeter of the orthogonal projection of the body onto a plane orthogonal to the assigned direction. Obvious examples show that there are pairs of convex bodies that have in every direction equal girths but are not translates of each other. Here a modification of the concept of the girth, called semigirth, is introduced, and it is shown that convex bodies are uniquely determined (up to translations) by their semi-girths. In addition, corresponding stability results are proved and an inequality concerning the central symmetry of convex domains is established.Let K be a convex body. In the present context that means a nonempty compact convex subset of euclidean three-dimensional space 3 . Let e denote the class of all planes in 3 that contain the origin o of 3 . For any E 2 e we write K E for the orthogonal projection of K onto E. The compact convex subsets of E will be referred to as convex domains, and the perimeter of a convex domain D will be denoted by pD. Then pK E is called the girth of K with respect to E. It is well-known that the girth of a convex body, considered as a function of E, does not uniquely determine the body. (Here and in the sequel, "uniquely" is to be understood as "uniquely up to translations".) A simple example that shows this lack of uniqueness is provided by comparing a ball of diameter d with a non-spherical convex body K of constant width d. On the other hand, the knowledge of the girth of a convex body does provide some information on the body itself. A striking result in this direction is the following theorem of Minkowski [12]: If the girth of a convex body K (viewed as a function on e ) is constant, then K has constant width. For further results and references concerning this and related results on convex bodies of constant width see the survey article [3]. A more general result than Minkowski's was proved by Nakajima [13]. To formulate it recall that the width of K in the direction u, denoted by w K u, is defined as the distance between the two parallel support planes of K that are orthogonal to u. Two convex bodies K and L are said to be equiwide if for every direction w K u w L u. Nakajima's result can be formulated as follows: If K and L are two convex bodies such that pK E pL E (for all E 2 e ), then K and L are equiwide. Actually the proof of this theorem requires only slight modifications of the original proof of Minkowski which depends on the expansion of the support function of a convex body as a series of spherical harmonics. Details and substantial generalizations of this

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Stability of Geometric Inequalities

Cited by 46 publications

References 30 publications

The Brunn-Minkowski inequality

The Brunn-Minkowski inequality

Stability of Blaschke's Characterization of Ellipsoids and Radon Norms

On the girth of convex bodies

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