Two Volume Product Inequalities and Their Applications

2009 ◽  
Vol 52 (3) ◽  
pp. 464-472 ◽  
Author(s):  
Alina Stancu
Keyword(s):  
Convex Body ◽  
Gauss Curvature ◽  
Floating Body ◽  
Volume Product ◽  
Positive Gauss Curvature

AbstractLet K ⊂ ℝn+1 be a convex body of class C2 with everywhere positive Gauss curvature. We show that there exists a positive number δ(K) such that for any δ ∈ (0, δ(K)) we have Vol(Kδ) · Vol((Kδ)*) ≥ Vol(K) · Vol(K*) ≥ Vol(Kδ) · Vol((Kδ)*), where Kδ, Kδ and K* stand for the convex floating body, the illumination body, and the polar of K, respectively. We derive a few consequences of these inequalities.

scholarly journals The geometry of a positively curved Zoll surface of revolution

2019 ◽  
Vol 16 (supp02) ◽  
pp. 1941003 ◽  
Author(s):  
Kazuyoshi Kiyohara ◽  
Sorin V. Sabau ◽  
Kazuhiro Shibuya
Keyword(s):  
Gauss Curvature ◽  
Flag Curvature ◽  
Finsler Metrics ◽  
Surface Of Revolution ◽  
Constant Flag Curvature ◽  
Explicit Constructions ◽  
Positively Curved ◽  
Positive Gauss Curvature

In this paper, we study the geometry of the manifolds of geodesics of a Zoll surface of positive Gauss curvature, show how these metrics induce Finsler metrics of constant flag curvature and give some explicit constructions.

scholarly journals Bartnik mass via vacuum extensions

2019 ◽  
Vol 30 (13) ◽  
pp. 1940006
Author(s):  
Pengzi Miao ◽  
Naqing Xie
Keyword(s):  
Initial Data ◽  
Scalar Curvature ◽  
Apparent Horizon ◽  
Gauss Curvature ◽  
Positive Mass Theorem ◽  
Data Sets ◽  
Penrose Inequality ◽  
Asymptotically Flat ◽  
Apparent Horizons ◽  
Positive Gauss Curvature

We construct asymptotically flat, scalar flat extensions of Bartnik data [Formula: see text], where [Formula: see text] is a metric of positive Gauss curvature on a two-sphere [Formula: see text], and [Formula: see text] is a function that is either positive or identically zero on [Formula: see text], such that the mass of the extension can be made arbitrarily close to the half area radius of [Formula: see text]. In the case of [Formula: see text], the result gives an analog of a theorem of Mantoulidis and Schoen [On the Bartnik mass of apparent horizons, Class. Quantum Grav. 32(20) (2015) 205002, 16 pp.], but with extensions that have vanishing scalar curvature. In the context of initial data sets in general relativity, the result produces asymptotically flat, time-symmetric, vacuum initial data with an apparent horizon [Formula: see text], for any metric [Formula: see text] with positive Gauss curvature, such that the mass of the initial data is arbitrarily close to the optimal value in the Riemannian Penrose inequality. The method we use is the Shi–Tam type metric construction from [Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature, J. Differential Geom. 62(1) (2002) 79–125] and a refined Shi–Tam monotonicity, found by the first named author in [On a localized Riemannian Penrose inequality, Commun. Math. Phys. 292(1) (2009) 271–284].

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

Gate coupling and floating-body effects in thin-film SOI MOSFETs

1988 ◽  
Vol 24 (4) ◽  
pp. 238 ◽  
Author(s):  
S.S. Tsao ◽  
D.R. Myers ◽  
G.K. Celler
Keyword(s):  
Thin Film ◽  
Floating Body ◽  
Floating Body Effects

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.

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