Closeness to spheres of hypersurfaces with normal curvature bounded below

Borisenko, Alexander A.; Drach, Kostiantyn

doi:10.1070/sm2013v204n11abeh004349

Cited by 9 publications

(17 citation statements)

References 9 publications

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“…One of the natural ways to do this is to consider curves of bounded curvature. Such class appeared in a number of extremal problems (see, for example, [1,5,6,10,11,13], and also [15]). In particular, in [10] the authors gave a bound on the area of domains enclosed by closed embedded plane curves of the fixed lengths whose curvatures k satisfy the inequality |k| 1 and the lengths satisfy some additional restrictions.…”

Section: Introductionmentioning

confidence: 99%

Extreme Properties of Curves with Bounded Curvature on a Sphere

Borisenko

Drach

2014

J Dyn Control Syst

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Section: Introductionmentioning

confidence: 99%

Extreme Properties of Curves with Bounded Curvature on a Sphere

Borisenko

Drach

2014

J Dyn Control Syst

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“…for some small open neighborhood U (p) ⊂ R n+1 of p (see [BDr1,Dr2]). Although λ-convexity and λ-concavity seem to be two notions dual to each other, methods and difficulties in solving the reverse isoperimetric problem in each of these classes are quite distinct.…”

Section: 2mentioning

confidence: 99%

“…At the same time, motivated by the study of strictly convex hypersurfaces in Riemannian spaces (see, for instance, [BM,Bor1,BDr1]), Borisenko and Drach in a series of papers [BDr2,BDr3,Dr1] obtained two-dimensional reverse isoperimetric inequalities for so-called λ-convex curves, i.e. curves whose curvature k, in a weak sense, satisfies k λ > 0.…”

Section: Introductionmentioning

confidence: 99%

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A sausage body is a unique solution for a reverse isoperimetric problem

Chernov

Drach

Tatarko

2019

Advances in Mathematics

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We consider the class of λ-concave bodies in R n+1 ; that is, convex bodies with the property that each of their boundary points supports a tangent ball of radius 1/λ that lies locally (around the boundary point) inside the body. In this class we solve a reverse isoperimetric problem: we show that the convex hull of two balls of radius 1/λ (a sausage body) is a unique volume minimizer among all λ-concave bodies of given surface area. This is in a surprising contrast to the standard isoperimetric problem for which, as it is well-known, the unique maximizer is a ball. We solve the reverse isoperimetric problem by proving a reverse quermassintegral inequality, the second main result of this paper.

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“…In this section we will prove Theorem 1 using the similar technique as in [6], but our proof will be shorter.…”

Section: Proof Of Theoremmentioning

confidence: 99%

Comparison theorem for the support functions of hypersurfaces

Borisenko¹,

Drach²

2015

Dopov. Nac. akad. nauk Ukr.

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Abstract. For a convex domain D that is enclosed by the hypersurface ∂D of bounded normal curvature, we prove an angle comparison theorem for angles between ∂D and geodesic rays starting from some fixed point in D, and the corresponding angles for hypersurfaces of constant normal curvature. Also, we obtain a comparison theorem for support functions of such surfaces. As a corollary, we present a proof of Blaschke's Rolling Theorem. Preliminaries and the main resultsIs it known the following theorem due to W. Blaschke:be a convex body with the C r -smooth boundary ∂D (r 2), and P ∈ ∂D be an arbitrary point. Let ∂D λ ⊂ M m (c) be a complete hypersurface of constant normal curvature equal to some λ > 0, and suppose that ∂D λ touches ∂D at P so that their inner unit normals coincide.A. If normal curvatures k n of the hypersurface ∂D at all points and in all directions satisfy the inequality k n λ, then ∂D lies entirely in the closed convex domain bounded by ∂D λ .B. If normal curvatures of the hypersurface ∂D at all points and in all directions satisfy the inequality λ k n , then the hypersurface ∂D λ lies in D.Moreover, the hypersurfaces ∂D and ∂D λ can intersect only by a domain that contains the point P .For the Euclidean space this theorem was first proved in [1]; for the general case of constant curvature spaces see [2,3,4].It appears that Blaschke's Rolling Theorem can be obtained as a corollary from the following comparison theorems for angles between the radius-vector of a hypersurface and its normals. In order to give exact statements, we need to agree on some notations.Everywhere below let M m be a complete simply-connected m-dimensional Riemannian manifold such that its sectional curvatures K σ in a direction of a 2-plane σ ⊂ T M m satisfy the inequality c 2 K σ c 1 with some constants c 1 and c 2 .2010 Mathematics Subject Classification. 53C20.

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Closeness to spheres of hypersurfaces with normal curvature bounded below

Cited by 9 publications

References 9 publications

Extreme Properties of Curves with Bounded Curvature on a Sphere

Extreme Properties of Curves with Bounded Curvature on a Sphere

A sausage body is a unique solution for a reverse isoperimetric problem

Comparison theorem for the support functions of hypersurfaces

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