The Four Vertex Theorem and its Converse

阿基米德球面定理的一个逆定理

A Converse to the Archimedes Sphere Theorem

本文证明若凸曲面被固定宽度平板截取的面积仅依赖方向，则该曲面为球面，推广了布拉施克和斯塔姆的定理，并适用于光滑闭曲面，证明结合牛顿势、静电刻画及拉东变换的解析性质。

Projective geometry of polygons and discrete 4-vertex and 6-vertex theorems

The paper concerns discrete versions of the three well-known results of projective differential geometry: the four vertex theorem, the six affine vertex theorem and the Ghys theorem on four zeroes of the Schwarzian derivative. We study geometry of closed polygonal lines in $\bbRP^d$ and prove that polygons satisfying a certain convexity condition have at least d+1 flattenings. This result provides a new approach to the above mentioned classical theorems.

Reflection quotients in Riemannian Geometry. A Geometric Converse to Chevalley's Theorem

Chevalley's theorem and it's converse, the Sheppard-Todd theorem, assert that finite reflection groups are distinguished by the fact that the ring of invariant polynomials is freely generated. We show that in the Euclidean case, a weaker condition suffices to characterize finite reflection groups, namely that a freely-generated polynomial subring is closed with respect to the gradient product.

Converse Sturm-Hurwitz-Kellogg theorem and related results

The classical Sturm-Hurwitz-Kellogg theorem asserts that a function, orthogonal to an n-dimensional Chebyshev system on a circle, has at least n+1 sign changes. We prove the converse: given an n-dimensional Chebyshev system on a circle and a function with at least n+1 sign changes, there exists an orientation preserving diffeomorphism of the circle that takes this function to a function, orthogonal to the Chebyshev system. We also prove that if a function on the real projective line has at least four sign changes then there exists an orientation preserving diffeomorphism of the projective line…

Moon in a puddle and the four-vertex theorem

We present a proof of the Moon in a puddle theorem, and use its key lemma to prove a generalization of the four-vertex theorem.

A partial converse to the Andreotti-Grauert theorem

Let $X$ be a smooth projective manifold with $\dim_\mathbb{C} X=n$. We show that if a line bundle $L$ is $(n-1)$-ample, then it is $(n-1)$-positive. This is a partial converse to the Andreotti-Grauert theorem. As an application, we show that a projective manifold $X$ is uniruled if and only if there exists a Hermitian metric $ω$ on $X$ such that its Ricci curvature $\mathrm{Ric}(ω)$ has at least one positive eigenvalue everywhere.