Continuity of the solution to the even Lp Minkowski problem for 0 < p < 1 in the plane

2020 ◽  
Vol 31 (12) ◽  
pp. 2050101
Author(s):  
Hejun Wang ◽  
Yusha Lv
Keyword(s):  
Weak Convergence ◽  
Surface Area ◽  
Hausdorff Metric ◽  
Convex Bodies ◽  
Minkowski Problem

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.

scholarly journals Inequalities for the Surface Area of Projections of Convex Bodies

2018 ◽  
Vol 70 (4) ◽  
pp. 804-823 ◽  
Author(s):  
Apostolos Giannopoulos ◽  
Alexander Koldobsky ◽  
Petros Valettas
Keyword(s):  
Convex Body ◽  
Surface Area ◽  
Convex Bodies ◽  
Maximal Surface ◽  
Projection Body ◽  
Projections Of Convex Bodies ◽  
Lower Dimensional

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.

scholarly journals Convex Bodies of Minimal Volume, Surface Area and Mean Width with Respect to Thin Shells

2008 ◽  
Vol 60 (1) ◽  
pp. 3-32 ◽  
Author(s):  
Károly Böröczky ◽  
Károly J. Böröczky ◽  
Carsten Schütt ◽  
Gergely Wintsche
Keyword(s):  
Unit Ball ◽  
Surface Area ◽  
Convex Bodies ◽  
Extreme Points ◽  
Thin Shells ◽  
Minimal Volume ◽  
Asymptotic Formulae ◽  
Mean Width

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.

Asymptotic Formulae for the Lattice Point Enumerator

1999 ◽  
Vol 51 (2) ◽  
pp. 225-249 ◽  
Author(s):  
U. Betke ◽  
K. Böröczky
Keyword(s):  
Convex Body ◽  
Surface Area ◽  
Positive Curvature ◽  
Convex Bodies ◽  
Mixed Volume ◽  
Lattice Points ◽  
Asymptotic Formulae ◽  
General Convex ◽  
Lattice Surface ◽  
Small Deformations

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.

scholarly journals A Class of Star-Shaped Bodies

1959 ◽  
Vol 2 (3) ◽  
pp. 175-180 ◽  
Author(s):  
Z.A. Melzak
Keyword(s):  
Surface Area ◽  
Lipschitz Condition ◽  
Convex Bodies ◽  
Nonempty Interior ◽  
Empty Interior ◽  
Selection Principle ◽  
Positive Functions ◽  
Bounded Sets ◽  
Bounded Convex

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.

scholarly journals General Blaschke Bodies and the Asymmetric Negative Solutions of Shephard Problem

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 610
Author(s):  
Tian Li ◽  
Weidong Wang ◽  
Yaping Mao
Keyword(s):  
Surface Area ◽  
Convex Bodies ◽  
Affine Surface ◽  
Affine Surface Area ◽  
Extremal Values ◽  
Shephard Problem ◽  
Projection Bodies

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.

Comparing the M-position with some classical positions of convex bodies

2011 ◽  
Vol 152 (1) ◽  
pp. 131-152 ◽  
Author(s):  
E. MARKESSINIS ◽  
G. PAOURIS ◽  
CH. SAROGLOU
Keyword(s):  
Minimal Surface ◽  
Surface Area ◽  
Convex Bodies ◽  
Mean Width ◽  
Quantitative Results ◽  
Minimal Surface Area

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.

scholarly journals Index of Grassmann manifolds and orthogonal shadows

2018 ◽  
Vol 30 (6) ◽  
pp. 1539-1572
Author(s):  
Djordje Baralić ◽  
Pavle V. M. Blagojević ◽  
Roman Karasev ◽  
Aleksandar Vučić
Keyword(s):  
Vector Bundle ◽  
Convex Body ◽  
Wreath Product ◽  
Hausdorff Metric ◽  
Convex Bodies ◽  
Orthogonal Complement ◽  
Grassmann Manifolds ◽  
The Real ◽  
Geometric Result

Abstract In this paper, we study the {\mathbb{Z}/2} action on the real Grassmann manifolds {G_{n}(\mathbb{R}^{2n})} and {\widetilde{G}_{n}(\mathbb{R}^{2n})} given by taking the (appropriately oriented) orthogonal complement. We completely evaluate the related {\mathbb{Z}/2} Fadell–Husseini index utilizing a novel computation of the Stiefel–Whitney classes of the wreath product of a vector bundle. These results are used to establish the following geometric result about the orthogonal shadows of a convex body: For {n=2^{a}(2b+1)} , {k=2^{a+1}-1} , a convex body C in {\mathbb{R}^{2n}} , and k real-valued functions {\alpha_{1},\ldots,\alpha_{k}} continuous on convex bodies in {\mathbb{R}^{2n}} with respect to the Hausdorff metric, there exists a subspace {V\subseteq\mathbb{R}^{2n}} such that projections of C to V and its orthogonal complement {V^{\perp}} have the same value with respect to each function {\alpha_{i}} , that is, {\alpha_{i}(p_{V}(C))=\alpha_{i}(p_{V^{\perp}}(C))} for all {1\leq i\leq k} .

Section means, integral transforms, and Boolean models

1996 ◽  
Vol 28 (2) ◽  
pp. 332-333
Author(s):  
Paul Goodey ◽  
Markus Kiderlen ◽  
Wolfgang Well
Keyword(s):  
Surface Area ◽  
Support Function ◽  
Convex Geometry ◽  
Convex Bodies ◽  
Integral Transforms ◽  
Boolean Models ◽  
Area Measure ◽  
Surface Area Measure ◽  
Left Hand ◽  
The Mean

For a stationary particle process X with convex particles in ℝdd ≧ 2, a mean body M(X) can be defined by where h(M,·) denotes the support function of the convex body M, γ the intensity of X, and P0 is the distribution of the typical particle of X (a probability measure on the set of convex bodies with Steiner point at the origin). Replacing the support function h(M,·) by the surface area measure S(M,·) (see Schneider (1993), for the basic notions from convex geometry), we get the Blaschke body B(X) of X, After normalization, the left-hand side represents the mean normal distribution of X. The main problem discussed here is whether B(X) (respectively S(B(X), ·)) is uniquely determined by the mean bodies M(X ∩ E) in random planar sections X ∩ E of X. From more general results in Weil (1995), it follows that the expectation ES(M(X ∩ E), ·) (taken w.r.t. the uniform distribution of two-dimensional subspaces E in ℝd) equals the surface area measure of a section mean B2(B(X)) of B(X). Thus, the formulated stereological question can be reduced to the injectivity of the transform B2 : K ↦ B2(K).

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