arXiv research

Improved agnostic learning time via Gaussian surface area analysis.

It is shown that $3$ disjoint sets with fixed Gaussian volumes that partition $\mathbb{R}^{n}$ with nearly minimum total Gaussian surface area must be close to adjacent $120$ degree sectors, when $n\geq2$. These same results hold for any number $m\leq n+1$ of sets partitioning $\mathbb{R}^{n}$, conditional on the solut…

Consider a random smooth Gaussian field $G(x):F\to\mathbb{R}$, where $F$ is a compact in $\mathbb{R}^d$. We derive a formula for average area of a surface generated by the equation $G(x)=0$ and give some applications. As an auxiliary result we obtain an integral expression for area of a surface induced by zeros of a \e…

Characterizes area-minimizing maps for surfaces of genus ≥ 2.

The paper proves that symmetric sets with minimal Gaussian surface area are nearly convex cylinders.

Study on discrete Gaussian curvature for polyhedral surfaces.

The normal map given by Birkhoff orthogonality yields extensions of principal, Gaussian and mean curvatures to surfaces immersed in three-dimensional spaces whose geometry is given by an arbitrary norm and which are also called Minkowski spaces. We obtain characterizations of the Minkowski Gaussian curvature in terms o…

The paper introduces a new discretization of Gaussian curvature on surfaces.

New weighted surface area measures for convex bodies with applications.

We consider a general theory of curvatures of discrete surfaces equipped with edgewise parallel Gauss images, and where mean and Gaussian curvatures of faces are derived from the faces' areas and mixed areas. Remarkably these notions are capable of unifying notable previously defined classes of surfaces, such as discre…

Let $Ω\subset\mathbb{R}^{n+1}$ have minimal Gaussian surface area among all sets satisfying $Ω=-Ω$ with fixed Gaussian volume. Let $A=A_{x}$ be the second fundamental form of $\partialΩ$ at $x$, i.e. $A$ is the matrix of first order partial derivatives of the unit normal vector at $x\in\partialΩ$. For any $x=(x_{1},\ld…

Non-negative $L_1$-approximating polynomials for Gaussian distributions are proven for certain classes of sets.

The paper proves a theorem for discretizing Gaussian curvature on surfaces.

Solves a long-standing convex geometry problem about mixed volumes.

We prove a generalization of the classical Gauss-Bonnet formula for a conical metric on a compact Riemann surface provided that the Gaussian curvature is Lebesgue integrable with respect to the area form of the metric. We also construct explicitly some conical metrics whose curvature is not integrable.

It is shown that $m$ disjoint sets with fixed Gaussian volumes that partition $\mathbb{R}^{n}$ with minimum Gaussian surface area must be $(m-1)$-dimensional. This follows from a second variation argument using infinitesimal translations. The special case $m=3$ proves the Double Bubble problem for the Gaussian measure,…

We study the problem of estimating the parameters of a Gaussian distribution when samples are only shown if they fall in some (unknown) subset $S \subseteq \R^d$. This core problem in truncated statistics has long history going back to Galton, Lee, Pearson and Fisher. Recent work by Daskalakis et al. (FOCS'18), provide…

We show the flexibility of the metric entropy and obtain additional restrictions on the topological entropy of geodesic flow on closed surfaces of negative Euler characteristic with smooth non-positively curved Riemannian metrics with fixed total area in a fixed conformal class. Moreover, we obtain a collar lemma, a th…

Let $M$ be a closed oriented surface of negative Gaussian curvature and let $Ω$ be a non-exact 2-form. Let $λ$ be a small positive real number. We show that the longitudinal KAM-cocycle of the magnetic flow given by $\la Ω$ is a coboundary if and only if the Gaussian curvature is constant and $Ω$ is a constant multiple…

Study sharp upper bounds for Aharonov-Bohm eigenvalues on surfaces.

Generalizes Gauss-Bonnet to metrics with logarithmic singularities.

For all $k\in]0,1[$, we construct a canonical bijection between the space of ramified coverings of the sphere and the space of complete immersed surfaces in $3$-dimensional hyperbolic space of finite area and of constant extrinsic curvature equal to $k$. We show, furthermore, that this bijection restricts to a homeomor…

In this paper, we introduce the $L_p$ geominimal surface area for all $-n\neq p<1$, which extends the classical geominimal surface area ($p=1$) by Petty and the $L_p$ geominimal surface area by Lutwak ($p>1$). Our extension of the $L_p$ geominimal surface area is motivated by recent work on the extension of the $L_p$ a…

In this paper, we generalize our results in \cite{GX3} to triangulated surfaces in hyperbolic background geometry, which means that all triangles can be embedded in the standard hyperbolic space. We introduce a new discrete Gaussian curvature by dividing the classical discrete Gauss curvature by an area element, which …

Overview of affine surface area and its history.

The paper connects least area surfaces to quasi-normal surfaces in 3-manifolds.

Two families of general affine surface areas are introduced. Basic properties and affine isoperimetric inequalities for these new affine surface areas as well as for $L_φ$ affine surface areas are established.

Study spherical convex bodies using $L_p$-floating areas and curvature entropy.

Introduces new weighted floating functions and affine surface areas.

The authors Balogh-Tyson-Vecchi in arXiv:1604.00180 utilize the Riemannian approximations scheme $(\mathbb H^1,<,>_L)$, in the Heisenberg group, introduced by Gromov, to calculate the limits of Gaussian and normal curvatures defined on surfaces of $\mathbb H^1$ when $L\rightarrow\infty$. They show that these limits exi…

Upper bounds on area for surfaces with constant mean curvature in hyperbolic 3-manifolds.

A proof that hyperbolic plane cannot be immersed in Euclidean 3-space.

We study an area minimization problem for spacelike zero mean curvature surfaces in four dimensional Lorentz-Minkowski space. The areas of these surfaces are compared of with the areas of certain marginally trapped surfaces having the same boundary values.

Study on existence and structure of P-area surfaces in Heisenberg group.

Minimal surfaces in hyperbolic space have a renormalized area criterion.

Classifies area-minimizing surfaces in R^4 as algebraic.

New surface area measures defined for ball-convex bodies, leading to entropy and inequalities.

Minimal surfaces in hyperbolic space have a sharp area bound.

We give a short and simple proof of Cauchy's surface area formula, which states that the average area of a projection of a convex body is equal to its surface area up to a multiplicative constant in the dimension.

Entropy is a natural geometric quantity measuring the complexity of a surface embedded in $\mathbb{R}^3$. For dynamical reasons relating to mean curvature flow, Colding-Ilmanen-Minicozzi-White conjectured that the entropy of any closed surface is at least that of the self-shrinking two-sphere. We prove this conjecture …

Minimal surfaces in a ball have limited area.

We study the classification of area-stationary and stable $C^2$ regular surfaces in the space of the rigid motions of the Minkowski plane E(1,1), equipped with its sub-Riemannian structure. We construct examples of area-stationary surfaces that are not foliated by sub-Riemannian geodesics. We also prove that there exis…

Generalized Cauchy's surface area formula to arbitrary submanifolds in R^n.

We use variational arguments to introduce a notion of mean curvature for surfaces in the Heisenberg group H^1 endowed with its Carnot-Carathéodory distance. By analyzing the first variation of area, we characterize C^2 stationary surfaces for the area as those with mean curvature zero (or constant if a volume-preservin…

New minimal surfaces grow area very quickly.

In this paper, we introduce several mixed $L_p$ geominimal surface areas for multiple convex bodies for all $p\neq -n$. Our definitions are motivated from an equivalent formula for the mixed $p$-affine surface area. Some properties, such as the affine invariance, for these mixed $L_p$ geominimal surface areas are prove…

The paper develops inequalities for log-concave functions and related surface areas.

New illumination bodies defined for ball-convex shapes, proving convexity and establishing surface area measures.