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 Lp−Winterniz problem on firey projection of convex bodies

2014 ◽  
Vol 45 (2) ◽  
pp. 179-193
Author(s):  
Tong Yi MA ◽  
Li Li Zhang
Keyword(s):  
Convex Body ◽  
Surface Area ◽  
Convex Bodies ◽  
Affine Surface ◽  
Projection Body ◽  
Affine Surface Area ◽  
Funk Transform

For $p\geq 1$, Lutwak, Yang and Zhang introduced the concept of $p$-projection body, and Lutwak introduced the concept of $L_{p}-$ affine surface area of convex body. In this paper, we develop the Minkowski-Funk transform approach in the $L_{p}$-Brunn-Minkowski theory. We consider the question of whether $\Pi_{p}K\subseteq \Pi_{p}L$ implies $\Omega_{p}(K) \leq \Omega_{p}(L)$, where $\Pi_{p}K$ and $\Omega_{p}K$ denotes the $p-$projection body of convex body $K$ and the $L_{p}-$affine surface area of convex body $K$, respectively. We also formulate and solve a generalized $L_{p}-$Winterniz problem for Firey projections.

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 Centrally symmetric convex bodies and sections having maximal quermassintegrals

2012 ◽  
Vol 49 (2) ◽  
pp. 189-199
Author(s):  
E. Makai ◽  
H. Martini
Keyword(s):  
Convex Body ◽  
Convex Bodies ◽  
The Body ◽  
Symmetric Convex ◽  
Local Theorem ◽  
Centrally Symmetric ◽  
Euclidean Unit Ball ◽  
Analogous Theorem ◽  
Lower Dimensional ◽  
Dimensional Surface

Let d ≧ 2, and let K ⊂ ℝd be a convex body containing the origin 0 in its interior. In a previous paper we have proved the following. The body K is 0-symmetric if and only if the following holds. For each ω ∈ Sd−1, we have that the (d − 1)-volume of the intersection of K and an arbitrary hyperplane, with normal ω, attains its maximum if the hyperplane contains 0. An analogous theorem, for 1-dimensional sections and 1-volumes, has been proved long ago by Hammer (see [2]). In this paper we deal with the ((d − 2)-dimensional) surface area, or with lower dimensional quermassintegrals of these intersections, and prove an analogous, but local theorem, for small C2-perturbations, or C3-perturbations of the Euclidean unit ball, respectively.

scholarly journals Almost ellipsoidal sections and projections of convex bodies

1975 ◽  
Vol 77 (3) ◽  
pp. 529-546 ◽  
Author(s):  
D. G. Larman ◽  
P. Mani
Keyword(s):  
Convex Body ◽  
High Dimension ◽  
Interior Point ◽  
Convex Bodies ◽  
Symmetric Convex Body ◽  
Symmetric Convex ◽  
Centrally Symmetric ◽  
Dimensional Section ◽  
Projections Of Convex Bodies

In (1) Dvoretsky proved, using very ingenious methods, that every centrally symmetric convex body of sufficiently high dimension contains a central k-dimensional section which is almost spherical. Here we shall extend this result (Corollary to Theorem 2) to k-dimensional sections through an arbitrary interior point of any convex body.

scholarly journals Uniform cover inequalities for the volume of coordinate sections and projections of convex bodies

2018 ◽  
Vol 18 (3) ◽  
pp. 345-354 ◽  
Author(s):  
Silouanos Brazitikos ◽  
Apostolos Giannopoulos ◽  
Dimitris-Marios Liakopoulos
Keyword(s):  
Convex Bodies ◽  
Upper Bounds ◽  
Compact Set ◽  
Uniform Cover ◽  
Cover Inequalities ◽  
Projections Of Convex Bodies ◽  
Lower Dimensional

AbstractThe classical Loomis–Whitney inequality and the uniform cover inequality of Bollobás and Thomason provide upper bounds for the volume of a compact set in terms of its lower dimensional coordinate projections. We provide further extensions of these inequalities in the setting of convex bodies. We also establish the corresponding dual inequalities for coordinate sections; these uniform cover inequalities for sections may be viewed as extensions of Meyer’s dual Loomis–Whitney inequality.

On Bodies Associated with a Given Convex Body

1996 ◽  
Vol 39 (4) ◽  
pp. 448-459 ◽  
Author(s):  
Endre Makai ◽  
Horst Martini
Keyword(s):  
Convex Body ◽  
Cross Section ◽  
Hausdorff Metric ◽  
Convex Bodies ◽  
Shadow Boundary ◽  
Projection Body ◽  
Intersection Body ◽  
Open Set ◽  
Difference Body ◽  
The Difference

AbstractLet d ≥ 2, and K ⊂ ℝd be a convex body with 0 ∈ int K. We consider the intersection body IK, the cross-section body CK and the projection body ΠK of K, which satisfy IK ⊂ CK ⊂ ΠK. We prove that [bd(IK)] ∩ [bd(CK)] ≠ (a joint observation with R. J. Gardner), while for d ≥ 3 the relation [CK] ⊂ int(ΠK) holds for K in a dense open set of convex bodies, in the Hausdorff metric. If IK = c ˙ CK for some constant c > 0, then K is centred, and if both IK and CK are centred balls, then K is a centred ball. If the chordal symmetral and the difference body of K are constant multiples of each other, then K is centred; if both are centred balls, then K is a centred ball. For d ≥ 3 we determine the minimal number of facets, and estimate the minimal number of vertices, of a convex d-polytope P having no plane shadow boundary with respect to parallel illumination (this property is related to the inclusion [CP] ⊂ int(ΠP)).

scholarly journals On the Monotonicity of the Isoperimetric Quotient for Parallel Bodies

2021 ◽  
Vol 32 (1) ◽  
Author(s):  
Christian Richter ◽  
Eugenia Saorín Gómez
Keyword(s):  
Convex Body ◽  
Surface Area ◽  
Convex Bodies ◽  
The Family ◽  
Definition Of ◽  
Euclidean Setting ◽  
Isoperimetric Quotient ◽  
Arbitrary Gauge ◽  
Area And Volume

AbstractThe isoperimetric quotient of the whole family of inner and outer parallel bodies of a convex body is shown to be decreasing in the parameter of definition of parallel bodies, along with a characterization of those convex bodies for which that quotient happens to be constant on some interval within its domain. This is obtained relative to arbitrary gauge bodies, having the classical Euclidean setting as a particular case. Similar results are established for different families of Wulff shapes that are closely related to parallel bodies. These give rise to solutions of isoperimetric-type problems. Furthermore, new results on the monotonicity of quotients of other quermassintegrals different from surface area and volume, for the family of parallel bodies, are obtained.

Measuring the Surface Area of a Convex Body

1944 ◽  
Vol 45 (4) ◽  
pp. 793 ◽  
Author(s):  
P. A. P. Moran
Keyword(s):  
Convex Body ◽  
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.

Prescribing Centro-Affine Curvature From One Convex Body to Another

2020 ◽  
Author(s):  
Alina Stancu
Keyword(s):  
Convex Body ◽  
Convex Bodies ◽  
Minkowski Inequality ◽  
Curvature Flow ◽  
Constant Volume ◽  
Convex Hypersurface ◽  
Strictly Convex ◽  
Initial Hypersurface ◽  
Affine Curvature ◽  
Convex Hypersurfaces

Abstract We study a curvature flow on smooth, closed, strictly convex hypersurfaces in $\mathbb{R}^n$, which commutes with the action of $SL(n)$. The flow shrinks the initial hypersurface to a point that, if rescaled to enclose a domain of constant volume, is a smooth, closed, strictly convex hypersurface in $\mathbb{R}^n$ with centro-affine curvature proportional, but not always equal, to the centro-affine curvature of a fixed hypersurface. We outline some consequences of this result for the geometry of convex bodies and the logarithmic Minkowski inequality.

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