Estimates of the principal radii of curvature of a closed convex hypersurface from the functions of its conditional radii of curvature

1991 ◽  
Vol 54 (1) ◽  
pp. 710-717
Author(s):  
V. N. Kokarev
Keyword(s):  
Convex Hypersurface ◽  
Closed Convex Hypersurface

scholarly journals New stability results for spheres and Wulff shapes

2018 ◽  
Vol 26 (2) ◽  
pp. 153-167 ◽  
Author(s):  
Julien Roth
Keyword(s):  
Euclidean Space ◽  
Convex Hypersurface ◽  
Wulff Shape ◽  
Space Forms ◽  
Closed Convex Hypersurface ◽  
Stability Results ◽  
Wulff Shapes

AbstractWe prove that a closed convex hypersurface of the Euclidean space with almost constant anisotropic first and second mean curvatures in the Lp-sense is W2,p-close (up to rescaling and translations) to the Wulff shape. We also obtain characterizations of geodesic hyperspheres of space forms improving those of [10] and [11].

Non-hyperbolic P-Invariant Closed Characteristics on Partially Symmetric Compact Convex Hypersurfaces

2018 ◽  
Vol 18 (4) ◽  
pp. 763-774
Author(s):  
Hui Liu ◽  
Gaosheng Zhu
Keyword(s):  
Compact Convex ◽  
Convex Hypersurface ◽  
Closed Characteristics ◽  
Compact Convex Hypersurfaces ◽  
Partially Symmetric ◽  
Convex Hypersurfaces

AbstractLet {n\geq 2} be an integer, {P=\mathrm{diag}(-I_{n-\kappa},I_{\kappa},-I_{n-\kappa},I_{\kappa})} for some integer {\kappa\in[0,n]}, and let {\Sigma\subset{\mathbb{R}}^{2n}} be a partially symmetric compact convex hypersurface, i.e., {x\in\Sigma} implies {Px\in\Sigma}, and {(r,R)}-pinched. In this paper, we prove that when {{R/r}<\sqrt{5/3}} and {0\leq\kappa\leq[\frac{n-1}{2}]}, there exist at least {E(\frac{n-2\kappa-1}{2})+E(\frac{n-2\kappa-1}{3})} non-hyperbolic P-invariant closed characteristics on Σ. In addition, when {{R/r}<\sqrt{3/2}}, {[\frac{n+1}{2}]\leq\kappa\leq n} and Σ carries exactly nP-invariant closed characteristics, then there exist at least {2E(\frac{2\kappa-n-1}{4})+E(\frac{n-\kappa-1}{3})} non-hyperbolic P-invariant closed characteristics on Σ, where the function {E(a)} is defined as {E(a)=\min{\{k\in{\mathbb{Z}}\mid k\geq a\}}} for any {a\in\mathbb{R}}.

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.

Locally Convex Hypersurfaces

1973 ◽  
Vol 25 (3) ◽  
pp. 531-538 ◽  
Author(s):  
L. B. Jonker ◽  
R. D. Norman
Keyword(s):  
Convex Body ◽  
Open Subset ◽  
Convex Hypersurface ◽  
Topological Manifold ◽  
Continuous Map ◽  
Open Set ◽  
Locally Convex ◽  
Convex Hypersurfaces

Let M be an n-dimensional connected topological manifold. Let ξ : M → Rn+1 be a continuous map with the following property: to each x ∈ M there is an open set x ∈ Ux ⊂ M, and a convex body Kx ⊂ Rn+1 such that ξ(UX) is an open subset of ∂Kx and such that is a homeomorphism onto its image. We shall call such a mapping ξ a locally convex immersion and, along with Van Heijenoort [8] we shall call ξ(M) a locally convex hypersurface of Rn+1.

Generalized Second Fundamental form for Lipschitzian Hypersurfaces by Way of Second Epi Derivatives

1992 ◽  
Vol 35 (4) ◽  
pp. 523-536 ◽  
Author(s):  
Dominikus Noll
Keyword(s):  
Fundamental Form ◽  
Second Fundamental Form ◽  
Convex Hypersurface ◽  
Parallel Surfaces ◽  
Limit Procedure

AbstractUsing second epi derivatives, we introduce a generalized second fundamental form for Lipschitzian hypersurfaces. In the case of a convex hypersurface, our approach leads back to the classical second fundamental form, which is usually obtained from the second fundamental forms of the outer parallel surfaces by means of a limit procedure.

scholarly journals On the affine diameter of closed convex hypersurfaces

2003 ◽  
Vol 68 (3) ◽  
pp. 431-437
Author(s):  
Weimin Sheng ◽  
Neil S. Trudinger
Keyword(s):  
Euclidean Space ◽  
Finite Volume ◽  
Convex Hypersurface ◽  
Uniformly Convex ◽  
Affine Diameter ◽  
Convex Hypersurfaces

In this paper we prove that the affine diameter of any closed uniformly convex hypersurface in Euclidean space enclosing finite volume is bounded from above.

Sign in / Sign up

Export Citation Format

Share Document