The Definition of Convex Surface - Mathematics Stack Exchange
The convex surface is a certain subset of the boundary of a convex set with non-empty interior. A convex set means every pair of points are joined by a line segment in the set.
Is there a way to relate convexity to Gaussian curvature?
Are there theorems relating Gaussian curvature of a surface to the convexity of the region of which it is the boundary? Is it at least true that the boundary surface of a convex subset of …
For a convex surface, why does the second fundamental form have …
Feb 13, 2023 · That’s precisely the point: For a surface, Gaussian curvature does not depend on the choice of the unit normal. However, changing that choice changes the sign of the second …
Pogorelov's rigidity theorem vs Cohn-Vossen rigidity theorem
Aug 9, 2019 · A convex surface is the boundary of a compact convex body in $\mathbb R^3$. Each convex surface comes with two metrics: the path-metric and the metric obtained by …
Intrinsic vs Extrinsic geometry of convex surfaces
Oct 16, 2017 · By Alexandrov's isometric embedding theorem, any locally convex metric prescribed on the sphere admits a realization as a convex surface in Euclidean 3-space, …
vector analysis - What exactly is meant by convex surfaces ...
Jun 23, 2019 · Google search didn't show up. It just shows up information related to spherical mirrors everywhere. Is there a way to intuitively (and maybe formally) define convex (and …
Immersed, locally (not globally) convex surfaces - MathOverflow
Jan 7, 2025 · 2 Locally convex complete immersed hypersurface in the Euclidean space is embedded and bounds a convex set. It was proved by James Stoker in his Über die Gestalt …
What is known about sufficient conditions for the rigidity of a …
Nov 5, 2018 · One direction in which one can generalize rigidity results for convex surfaces is rigidity of hyperbolic manifolds with convex boundary. (Indeed, rigidity of a convex surface …
Are there convex surfaces with negative Gaussian curvature?
Even so, wouldn't the saddle shape that the surface would take on near the point with negative curvature violate convexity? I have difficulty seeing how any convex surface could have a point …
Curved edges for a smooth three-dimensional object
Feb 22, 2023 · I assume we're talking about a 3D surface that doesn't have a boundary. In both of your references you state "A smooth surface has a well-defined tangent plane at every point of …