scholarly journals A Class of Star-Shaped Bodies

1959 ◽  
Vol 2 (3) ◽  
pp. 175-180 ◽  
Author(s):  
Z.A. Melzak

The more important properties of the class κ of all bounded convex bodies in E3 with non-empty interior include: uniform approximability by polyhedra, existence of volume and surface area, and Blaschke's selection principle, [l ], [2 ]. In this note we define and consider a class ℋ of star-shaped bodies in E3, which enjoys many properties of κ, among them the above-mentioned ones, and is considerably larger. Roughly speaking, ℋ consists of closed bounded sets in E3 with nonempty interior, whose boundary is completely visible from every point of a set with non-empty interior. It turns out that ℋ is identifiable with the class of all real-valued positive functions on the sphere S3 which satisfy a Lipschitz condition.

2020 ◽  
Vol 26 (1) ◽  
pp. 67-77 ◽  
Author(s):  
Silvestru Sever Dragomir

AbstractIn this paper, by the use of the divergence theorem, we establish some integral inequalities of Hermite–Hadamard type for convex functions of several variables defined on closed and bounded convex bodies in the Euclidean space {\mathbb{R}^{n}} for any {n\geq 2}.


2018 ◽  
Vol 70 (4) ◽  
pp. 804-823 ◽  
Author(s):  
Apostolos Giannopoulos ◽  
Alexander Koldobsky ◽  
Petros Valettas

AbstractWe provide general inequalities that compare the surface area S(K) of a convex body K in ℝn to the minimal, average, or maximal surface area of its hyperplane or lower dimensional projections. We discuss the same questions for all the quermassintegrals of K. We examine separately the dependence of the constants on the dimension in the case where K is in some of the classical positions or K is a projection body. Our results are in the spirit of the hyperplane problem, with sections replaced by projections and volume by surface area.


2008 ◽  
Vol 60 (1) ◽  
pp. 3-32 ◽  
Author(s):  
Károly Böröczky ◽  
Károly J. Böröczky ◽  
Carsten Schütt ◽  
Gergely Wintsche

AbstractGiven r > 1, we consider convex bodies in En which contain a fixed unit ball, and whose extreme points are of distance at least r from the centre of the unit ball, and we investigate how well these convex bodies approximate the unit ball in terms of volume, surface area and mean width. As r tends to one, we prove asymptotic formulae for the error of the approximation, and provide good estimates on the involved constants depending on the dimension.


2020 ◽  
Vol 31 (12) ◽  
pp. 2050101
Author(s):  
Hejun Wang ◽  
Yusha Lv

This paper concerns the continuity of the solution to the even [Formula: see text] Minkowski problem in the plane. When [Formula: see text], it is proved that the weak convergence of the even [Formula: see text] surface area measures implies the convergence of the corresponding convex bodies in the Hausdorff metric. Moreover, the continuity of the solution to the even [Formula: see text] Minkowski problem with respect to [Formula: see text] is also obtained.


1999 ◽  
Vol 51 (2) ◽  
pp. 225-249 ◽  
Author(s):  
U. Betke ◽  
K. Böröczky

AbstractLet M be a convex body such that the boundary has positive curvature. Then by a well developed theory dating back to Landau and Hlawka for large λ the number of lattice points in λM is given by G(λM) = V(λM) + O(λd−1−ε(d)) for some positive ε(d). Here we give for general convex bodies the weaker estimatewhere SZd (M) denotes the lattice surface area of M. The term SZd is optimal for all convex bodies and o(λd−1) cannot be improved in general. We prove that the same estimate even holds if we allow small deformations of M.Further we deal with families {Pλ} of convex bodies where the only condition is that the inradius tends to infinity. Here we havewhere the convex body K satisfies some simple condition, V(Pλ; K; 1) is some mixed volume and S(Pλ) is the surface area of Pλ.


Author(s):  
S.N. Melikhov ◽  
S. Momm

Let Qbe a bounded, convex, locally closed subset of CN with nonempty interior. For N>1 sufficient conditions are obtained that an operator of the representation of analytic functions on Q by exponential series has a continuous linear right inverse. For N=1 the criterions for the existence of a continuous linear right inverse for the representation operator are proved.


Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 610
Author(s):  
Tian Li ◽  
Weidong Wang ◽  
Yaping Mao

In this article, based on the Blaschke combination of convex bodies, we define the general Blaschke bodies and obtain the extremal values of their volume and affine surface area. Further, we study the asymmetric negative solutions of the Shephard problem for the projection bodies.


2011 ◽  
Vol 152 (1) ◽  
pp. 131-152 ◽  
Author(s):  
E. MARKESSINIS ◽  
G. PAOURIS ◽  
CH. SAROGLOU

AbstractThe purpose of this paper is to compare some classical positions of convex bodies. We provide exact quantitative results which show that the minimal surface area position and the minimal mean width position are not necessarily M-positions. We also construct examples of unconditional convex bodies of minimal surface area that exhibit the worst possible behavior with respect to their mean width or their minimal hyperplane projection.


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