On the inner parallel body of a convex body

1974 ◽  
Vol 19 (3) ◽  
pp. 217-219 ◽  
Author(s):  
P. McMullen
Keyword(s):  
Convex Body ◽  
Inner Parallel Body ◽  
Parallel Body

scholarly journals On the estimation of the medial axis and inner parallel body

2014 ◽  
Vol 129 ◽  
pp. 171-185
Author(s):  
Antonio Cuevas ◽  
Pamela Llop ◽  
Beatriz Pateiro-López
Keyword(s):  
Medial Axis ◽  
Inner Parallel Body ◽  
Parallel Body

Asymptotic Shape of Finite Packings

1998 ◽  
Vol 50 (1) ◽  
pp. 16-28 ◽  
Author(s):  
KáRoly Böröczky ◽  
Uwe Schnell
Keyword(s):  
Convex Body ◽  
Minimal Volume ◽  
Parametric Density ◽  
Asymptotic Shape ◽  
Parallel Body ◽  
Finite Packings

AbstractLet Kbe a convex body in Ed and denote by Cn the set of centroids of n non-overlapping translates of K. Forϱ > 0, assume that the parallel body conv Cn+ϱ K of convCn has minimal volume. The notion of parametric density (see [21]) provides a bridge between finite and infinite packings (see [4] or [14]). It is known that there exists a maximal ϱs(K) ≥ 1/(32d2) such that convCn is a segment for ϱ < ϱs(see [5]). We prove the existence of a minimal ϱc(K) ≤ d+ 1 such that if ϱ > ϱc and n is large then the shape of conv Cn can not be too far from the shape of K. For d= 2, we verify that ϱs= ϱc. For d≥ 3, we present the first example of a convex body with known ϱs and ϱc; namely, we have ϱs= ϱ c= 1 for the parallelotope.

scholarly journals On mean curvature integrals of the outer parallel body of the projection of a convex body

2014 ◽  
Vol 2014 (1) ◽  
Author(s):  
Chunna Zeng ◽  
Lei Ma ◽  
Yunwei Xia
Keyword(s):  
Convex Body ◽  
Mean Curvature ◽  
Parallel Body

Further inequalities and properties of p-inner parallel bodies

2020 ◽  
pp. 1-13
Author(s):  
Yingying Lou ◽  
Dongmeng Xi ◽  
Zhenbing Zeng
Keyword(s):  
Lower Bounds ◽  
Necessary Conditions ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Equality Case ◽  
Sufficient And Necessary Conditions ◽  
Inner Parallel Body ◽  
Parallel Body ◽  
Main Inequality

Abstract A. R. Martínez Fernández obtained upper bounds for quermassintegrals of the p-inner parallel bodies: an extension of the classical inner parallel body to the $L_p$ -Brunn-Minkowski theory. In this paper, we establish (sharp) upper and lower bounds for quermassintegrals of p-inner parallel bodies. Moreover, the sufficient and necessary conditions of the equality case for the main inequality are obtained, which characterize the so-called tangential bodies.

Equation of state for hard convex body fluid mixtures

1998 ◽  
Vol 94 (5) ◽  
pp. 809-814 ◽  
Author(s):  
C. BARRIO ◽  
J.R. SOLANA
Keyword(s):  
Equation Of State ◽  
Convex Body ◽  
Body Fluid ◽  
Fluid Mixtures

Generic rotation sets

2020 ◽  
pp. 1-13
Author(s):  
SEBASTIÁN PAVEZ-MOLINA
Keyword(s):  
Dynamical System ◽  
Convex Body ◽  
Probability Measures ◽  
Topological Dynamical System ◽  
Continuous Vector ◽  
Rotation Set ◽  
Vector Valued Function ◽  
Vector Valued ◽  
Uniquely Ergodic ◽  
Dense Set

Abstract Let $(X,T)$ be a topological dynamical system. Given a continuous vector-valued function $F \in C(X, \mathbb {R}^{d})$ called a potential, we define its rotation set $R(F)$ as the set of integrals of F with respect to all T-invariant probability measures, which is a convex body of $\mathbb {R}^{d}$ . In this paper we study the geometry of rotation sets. We prove that if T is a non-uniquely ergodic topological dynamical system with a dense set of periodic measures, then the map $R(\cdot )$ is open with respect to the uniform topologies. As a consequence, we obtain that the rotation set of a generic potential is strictly convex and has $C^{1}$ boundary. Furthermore, we prove that the map $R(\cdot )$ is surjective, extending a result of Kucherenko and Wolf.

scholarly journals Bounds on the Lattice Point Enumerator via Slices and Projections

2021 ◽  
Author(s):  
Ansgar Freyer ◽  
Martin Henk
Keyword(s):  
Convex Body ◽  
Lattice Point ◽  
Geometric Mean ◽  
Discrete Analogue ◽  
Lattice Points ◽  
General Context ◽  
General Bound

AbstractGardner et al. posed the problem to find a discrete analogue of Meyer’s inequality bounding from below the volume of a convex body by the geometric mean of the volumes of its slices with the coordinate hyperplanes. Motivated by this problem, for which we provide a first general bound, we study in a more general context the question of bounding the number of lattice points of a convex body in terms of slices, as well as projections.

An extremal property of the hypersphere

1951 ◽  
Vol 47 (1) ◽  
pp. 245-247 ◽  
Author(s):  
A. M. Macbeath
Keyword(s):  
Convex Body ◽  
Extremal Property ◽  
Plane Case ◽  
Simple Application ◽  
Plane Convex ◽  
Plane Convex Body

It was shown by Sas (1) that, if K is a plane convex body, then it is possible to inscribe in K a convex n-gon occupying no less a fraction of its area than the regular n-gon occupies in its circumscribing circle. It is the object of this note to establish the n-dimensional analogue of Sas's result, giving incidentally an independent proof of the plane case. The proof is a simple application of the Steiner method of symmetrization.

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
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