arXiv 1802.091922018-02math.DG

Convexity of the distance function to convex subsets of Riemannian manifolds

S. Khajehpour, M. R. Pouryayevali

英文摘要：

A characterization of the proximal normal cone is obtained and a separation theorem for convex subsets of Riemannian manifolds is established. Moreover, the convexity of the distance function $d_S$ for a convex subset $S$ in the cases where the boundary of $S$ contains a geodesic segment, the boundary of $S$ is $C^2$ or the boundary of $S$ is not regular is discussed. Furthermore, a nonsmooth version of positive semi-definiteness of Hessian of convex functions on Riemannian manifolds is established.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Geodesic Convexity Types in Riemannian Manifolds

The usual notion of set-convexity, valid in the classical Euclidean context, metamorphoses into several distinct convexity types in the more general Riemannian setting. By studying this phenomenon in reverse, we characterize complete manifolds for which certain convexity types are assumed a priori to coincide.

2016-11-26math.DG

半黎曼流形中超曲面的无穷小性和局部凸性

Infinitesimal and local convexity of a hypersurface in a semi-Riemannian manifold

给定一个黎曼流形M和M中的一个超曲面H，已知H中一点邻域上的无穷小凸蕴涵局部凸。本文证明了同样的结果在半黎曼流形中也成立。我们对只涉及类时测地线、零测地线或类空测地线的情形作了一些说明。几何凸性的概念也进行了审查，并给出了一些应用的洛伦兹流形的开子集的测地连通性。

2012-04-19math.DG

The convexity radius of a Riemannian manifold

The ratio of convexity radius over injectivity radius may be made arbitrarily small within the class of compact Riemannian manifolds of any fixed dimension at least two. This is proved using Gulliver's method of constructing manifolds with focal points but no conjugate points. The approach is suggested by a characterization of the convexity radius that resembles a classical result of Klingenberg about the injectivity radius.

2017-03-05math.DG

Local Estimate on Convexity Radius and decay of injectivity radius in a Riemannian manifold

In this paper we prove the following pointwise and curvature-free estimates on convexity radius, injectivity radius and local behavior of geodesics in a complete Riemannian manifold $M$: 1) the convexity radius of $p$, $\operatorname{conv}(p)\ge \min\{\frac{1}{2}\operatorname{inj}(p),\operatorname{foc}(B_{\operatorname{inj}(p)}(p))\}$, where $\operatorname{inj}(p)$ is the injectivity radius of $p$ and $\operatorname{foc}(B_r(p))$ is the focal radius of open ball centered at $p$ with radius $r$; 2) for any two points $p,q$ in $M$, $\operatorname{inj}(q)\ge \min\{\operatorname{inj}(p), \operator…

2017-06-29math.DG

关于非焦点流形上内射域的凸性

On the convexity of injectivity domains on nonfocal manifolds

对于光滑的非焦点紧致黎曼流形，我们证明了所谓的Ma-Trudinger-Wang条件蕴涵了内射性区域的凸性.这改进了Loeper和Villani先前的结果。

2014-04-01math.DG

Pluri-subharmonic functions on complex tori, Ricci curvature and convexity

We show that, in toric manifolds, one can characterize the sign of the Ricci curvature in terms of the convexity of the volume functional. More generally we discuss relationships between (i) Ricci curvature and volume, (ii) totally real and Lagrangian submanifolds, (iii) pluri-subharmonic functions and convexity.

2021-06-27math.DG

更多相关论文收起更多论文

平均曲率流的平均凸渐近锥形自膨胀子的凸性

Convexity of mean convex asymptotically conical self-expanders to the mean curvature flow

UCSB（加州大学圣塔芭芭拉分校）

针对平均曲率流中的平均凸渐近锥形自膨胀子，论文证明了当 $n\geq 3$ 且渐近锥平均凸、弱凸时，该完备自膨胀子必为严格凸。

2026-10-21math.DG

黎曼流形域的凸性

Convexity of domains of Riemannian manifolds

本文研究了黎曼流形子集的测地连通性问题。利用变分方法，证明了光滑黎曼流形上开域（其边界不可微且非凸）的测地连通性。在某些情况下，还获得了域的凸性。此外，我们提出的假设的适用性和独立性的例子。最后给出了动力系统具有固定能量轨线存在性的一个应用。

2000-04-12math.DGmath.DS

Convexity estimates for hypersurfaces moving by concave curvature functions

We study fully nonlinear geometric flows that deform strictly $k$-convex hypersurfaces in Euclidean space with pointwise normal speed given by a concave function of the principal curvatures. Specifically, the speeds we consider are obtained by performing a nonlinear interpolation between the mean and the $k$-harmonic mean of the principal curvatures. Our main result is a convexity estimate showing that, on compact solutions, regions of high curvature are approximately convex. In contrast to the mean curvature flow, the fully nonlinear flows considered here preserve $k$-convexity in a Riemannia…

2020-07-15math.DG

H-凸黎曼子流形

H-Convex Riemannian Submanifolds

在[4]、[5]中，许多作者利用欧氏凸超曲面的模型和[1]中的思想，定义并研究了黎曼流形的凸超曲面。正如第一作者在[7]中所证明的，超曲面的凸性与高斯曲率之间存在相互依赖关系。本文定义了任意余维黎曼子流形的$H$ -凸性，用中曲率向量$H$代替超曲面的法向量. B. Y. Chen [2]，[3]作为严格$H$ -凸性的定义，得到了它.

2005-11-01math.DGmath.OC

Convexity of Level Sets of Minimal Graph on Space Form with Nonnegative Curvature

For the minimal graph defined on a convex ring in the space form with nonnegative curvature, we obtain the regularity and the strict convexity about its level sets by the continuity method.

2016-07-20math.APmath.DG

p-凸性、p-多重次调和性与Levi问题

p-convexity, p-plurisubharmonicity and the Levi problem

建立了p-凸几何中的三个结果。首先是多复变数的Levi问题的类似问题，即：局部p-凸蕴涵全局p-凸。第二个断言，最小的p维电流的支持包含在与空间的“核心”的边界并集的p壳中。最后给出了p-正定矩阵凸锥中的极射线的特征。这是具有许多应用的基本结果。

2013-03-20math.APmath.DG

Geodesic Convex Optimization: Differentiation on Manifolds, Geodesics, and Convexity

EPFL（洛桑联邦理工学院）

Convex optimization is a vibrant and successful area due to the existence of a variety of efficient algorithms that leverage the rich structure provided by convexity. Convexity of a smooth set or a function in a Euclidean space is defined by how it interacts with the standard differential structure in this space -- the Hessian of a convex function has to be positive semi-definite everywhere. However, in recent years, there is a growing demand to understand non-convexity and develop computational methods to optimize non-convex functions. Intriguingly, there is a type of non-convexity that disap…

2018-06-17cs.DScs.LGmath.DG