arXiv research

The study connects polygon areas and projective structures in 3D space.

Study on Santaló point for convex bodies in normed spaces.

New optimal isosystolic inequality found for Finsler reversible 2-tori.

We prove that every Riemannian metric on the 2-disc such that all its geodesics are minimal, is a minimal filling of its boundary (within the class of fillings homeomorphic to the disc). This improves an earlier result of the author by removing the assumption that the boundary is convex. More generally, we prove this r…

The minimal area of Finsler disks with minimizing geodesics is at least 6/π r^2.

This paper is a continuation of the second author's previous work. We investigate the isoperimetric problem in the 2-dimensional Finsler space form $(F_B, B^2(1))$ with $k=0$ by using the Holmes-Thompson area and prove that the circle centered the origin achieves the local maximum area of the isoperimetric problem.

It is shown that if the Holmes-Thompson volume definition is used, totally geodesic submanifolds of a Finsler space are minimal. The analogous result for the Hausdorff measure is known to be false. ----- Nous montrons que les sous-varietes totalement geodesiques d'une variete de Finsler sont minimales pour le volume de…

We construct a class of Finsler metrics in three-dimensional space such that all their geodesics are lines, but not all planes are extremal for their Hausdorff area functionals. This shows that if the Hausdorff measure is used as notion of volume on Finsler spaces, then totally geodesic submanifolds are not necessarily…

The Funk metric connects billiards, projective geometry, and convex geometry.

We show that a non-compact (forward) complete Finsler manifold whose Holmes- Thompson volume is infinite admits no non-trivial convex functions. We apply this result to some Finsler manifolds whose Busemann function is convex.

This paper states a formula for the difference of the Holmes-Thompson volumes of two simple Finsler manifolds of arbitrary dimension, in terms of the boundary distances and their derivatives. An application is a preconditioned filling minimality result.

The paper proposes extensions of the usual notions of Finslerian volume to time orientable Finsler spacetime manifolds. The basic idea is to replace, in the classical Busemann-Hausdorff and Holmes-Thompson definitions, integration on the indicatrices of the given metric (which are, in Lorentzian signature, non-compact,…

We prove optimal systolic inequalities on Finsler Mobius bands relating the systole and the height of the Mobius band to its Holmes-Thompson volume. We also establish an optimal systolic in- equality for Finsler Klein bottles of revolution, which we conjecture to hold true for arbitrary Finsler metrics. Extremal metric…

The paper proposes a notion of volume element for Finsler spaces with metrics of Lorentzian signature, equipped with a time orientation. This notion is based on a slight modification of the idea of Holmes-Thompson volume element working for positive definite Finsler metrics and can be used in field-theoretical applicat…

New Finsler metric on sphere disproves systolic ratio conjecture.

Study on Funk geometry volume growth and polytope flags, verifying conjectures.

The contribution of this paper is two-fold. The first one is to derive a simple formula of the mean curvature form for a hypersurface in the Randers space with a Killing field, by considering the Busemann-Hausdorff measure and Holmes-Thompson measure simultaneously. The second one is to obtain the explicit local expres…

We show that the volume of a simple Riemannian metric on $D^n$ is locally monotone with respect to its boundary distance function. Namely if $g$ is a simple metric on $D^n$ and $g'$ is sufficiently close to $g$ and induces boundary distances greater or equal to those of $g$, then $vol(D^n,g')\ge vol(D^n,g)$. Furthermor…

Any Riemannian manifold has a canonical collection of valuations (finitely additive measures) attached to it, known as the intrinsic volumes or Lipschitz-Killing valuations. They date back to the remarkable discovery of H. Weyl that the coefficients of the tube volume polynomial are intrinsic invariants of the metric. …

In this note we introduce a natural Finsler structure on convex surfaces, referred to as the projective Finsler structure, which is dual in a sense to the obvious inclusion of a convex surface in a normed space. It has an associated projective girth, which is similar to the notion of girth defined by Schäffer. We prove…

In the first part of this dissertation, we give a new definition of a Laplace operator for Finsler metric as an average, with regard to an angle measure, of the second directional derivatives. This operator is elliptic, symmetric with respect to the Holmes-Thompson volume, and coincides with the usual Laplace--Beltrami…

NeuroPaint infers missing brain area dynamics from multi-animal datasets.

Develops calculus for random submanifolds using zonoids.

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…

Overview of affine surface area and its history.

Hasse principle applied to area-minimizing submanifolds across different homology types.

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.

The hermitian analog of Aleksandrov's area measures of convex bodies is investigated. A characterization of those area measures which arise as the first variation of unitarily invariant valuations is established. General smooth area measures are shown to form a module over smooth valuations and the module of unitarily …

BiLipschitz mappings can be extended to preserve area.

Formula for Heisenberg group surface areas derived.

Introduces new weighted floating functions and affine surface areas.

Study minimal annuli in a slab, estimating their area.

Study area-minimizing subgraphs in integer lattices.

Existing attention mechanisms are trained to attend to individual items in a collection (the memory) with a predefined, fixed granularity, e.g., a word token or an image grid. We propose area attention: a way to attend to areas in the memory, where each area contains a group of items that are structurally adjacent, e.g…

Minimal surfaces in hyperbolic space have a renormalized area criterion.

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

Study shows area-minimizing submanifolds are mostly rough, not smooth.

Study examines Hilbert area of inscribed polygons in projective geometry.

Minimal area of spun trefoil knot is found in 4D cubical space.

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.

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.

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

Pedal curves derived from ellipses are invariant in area.

We prove the existence of a continuous $BV$ minimizer with $C^{0}$ boundary value for the $p$-area (pseudohermitian or horizontal area) in a parabolically convex bounded domain. We extend the domain of the area functional from $BV$ functions to vector-valued measures. Our main purpose is to study the first and second v…

Contracts for Difference (CfDs) are forwards on the spread between an area price and the system price. Together with the system price forwards, these products are used to hedge the area price risk in the Nordic electricity market. The CfDs are typically available for the next two months, three quarters and three years.…

Random forests and LASSO methods improve small area estimation using auxiliary data.

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

Study area minimizing currents in conformal cones, solving Dirichlet problems.