Inequalities between volume, center of mass, circumscribed radius, order, and mean curvature

Inequalities between volume, center of mass, circumscribed radius, order, and mean curvature

By applying the spectral decomposition of a submanifold of a Euclidean space, we derive several sharp geometric inequalities which provide us some best possible relations between volume, center of mass, circumscribed radius, inscribed radius, order, and mean curvature of the submanifold. Several of our results sharpen some well-known geometric inequalities.

Differences of the Curvature Radius and the Curvature Center between Acetabulum and Femoral Head in Dysplastic Hip Joint

Mass conserved Allen-Cahn equation and volume preserving mean curvature flow

We consider a mass conserved Allen-Cahn equation ut = ∆u + ε−2(f(u)− ελ(t)) in a bounded domain with no flux boundary condition, where ελ(t) is the average of f(u(·, t)) and −f is the derivative of a double equal well potential. Given a smooth hypersurface γ0 contained in the domain, we show that the solution uε with appropriate initial data approaches, as ε ց 0, to a limit which takes only two...

Mean Curvature and Asymptotic Volume of Small Balls

where Bp(t) is the ball of radius t centered at p and Bp (t) is the portion of the ball lying inside the sphere SR . Our goal is to show that, up to a negligible term, a similar formula holds for any hypersurface S in Rn, the factor 1/R being replaced with the mean curvature of the hypersurface. We briefly recall what mean curvature is. Let S be a hypersurface of class C2 in Euclidean n-space R...

Correlation between Corneal Radius of Curvature and Corneal Eccentricity

Aim: The primary aim of this study was to find if there is any correlation between the corneal radius of curvature and its eccentricity. Method: 45 subjects participated in this study, 24 emmetropes, 18 myopes and three hyperopes. All subjects were free of ocular abnormalities and had no media opacities. All the subjects had normal ocular health and good visual acuity of 1.0 or better for both ...