arXiv research

Proves stability of cone-volume measure with nearly constant density.

The cone-volume measure of a polytope with centroid at the origin is proved to satisfy the subspace concentration condition. As a consequence a conjectured (a dozen years ago) fundamental sharp affine isoperimetric inequality for the U-functional is completely established -- along with its equality conditions.

We show that the cone-volume measure of a convex body with centroid at the origin satisfies the subspace concentration condition. This implies, among others, a conjectured best possible inequality for the $\mathrm{U}$-functional of a convex body. For both results we provide stronger versions in the sense of stability i…

This paper proves a curvature entropy inequality for non-symmetric convex bodies.

Solves a discrete logarithmic Minkowski problem for electrostatic p-capacity.

Study on higher-dimensional quasigeodesics in metric spaces.

Study finds volume minimization principle for conical Calabi-Yau structures on horospherical cones.

Paper solves a new Minkowski problem for a specific type of rigidity.

New insights from centro-affine geometry solve a key geometric conjecture.

Study invariant measures on measured laminations for subgroups of mapping class group.

New set-valued star-shaped risk measures introduced for better risk assessment.

The Bergman measure converges to the Zhang measure on a hybrid space.

Bayesian approach to robust risk measures under model uncertainty.

Introduces Star-Shaped deviation measures for risk analysis.

The paper studies dynamic star-shaped risk measures and their representation.

Transformers can interpolate between arbitrary measures.

Classifies invariant measures on specific character varieties.

Paper characterizes star-shaped risk measures and their properties.

Paper introduces quasi-logconvex risk measures and their properties.

Submodularity is studied for convex risk measures, including Expected Shortfall.

New geometric measure simplifies complex analysis.

The paper explores non-convex risk measures and their characterizations.

The paper calculates extreme measures in continuous time conic finance.

One often finds in the literature connections between measures of fairness and measures of feature importance employed to interpret trained classifiers. However, there seems to be no study that compares fairness measures and feature importance measures. In this paper we propose ways to evaluate and compare such measure…

Paper characterizes monotonic mean-deviation risk measures.

Introduces factor risk measures to assess risk relative to multiple factors.

Dual representations for robust risk measures and uncertainty sets.

A scalable approach to learning from probability measures using quantization.

Study on measurable pseudo-Anosov maps on surfaces.

In this paper, we propose a new method of Bayesian measurement for spectral deconvolution, which regresses spectral data into the sum of unimodal basis function such as Gaussian or Lorentzian functions. Bayesian measurement is a framework for considering not only the target physical model but also the measurement model…

Risk measures for multivariate financial positions are studied in a utility-based framework. Under a certain incomplete preference relation, shortfall and divergence risk measures are defined as the optimal values of specific set minimization problems. The dual relationship between these two classes of multivariate ris…

The Cannon-Thurston map's pushed measures on the circle are singular with respect to sphere measures.

New weighted surface area measures for convex bodies with applications.

Standardized fairness measures for continuous risk scores using Wasserstein distance.

Starting with the work of Preiss on the geometry of measures, the classification of uniform measures in $\mathbb R^d$ has remained open, except for $d=1$ and for compactly supported measures in $d=2$, and for codimension $1$. In this paper we study $1$-dimensional measures in $\mathbb R^d$ for all $d$ and classify unif…

Study SRB measures for Anosov actions on manifolds.

We study generalizations of Reifenberg's Theorem for measures in $\mathbb R^n$ under assumptions on the Jones' $β$-numbers, which appropriately measure how close the support is to being contained in a subspace. Our main results, which holds for general measures without density assumptions, give effective measure bounds…

Study proposes worst+gap measure for better DG evaluation.

The study evaluates AI model performance measures for medical use.

We introduce a weak notion of barycenter of a probability measure $μ$ on a metric measure space $(X, d, {\bf m})$, with the metric $d$ and reference measure ${\bf m}$. Under the assumption that optimal transport plans are given by mappings, we prove that our barycenter $B(μ)$ is well defined; it is a probability measur…

Study on convergence of Narasimhan-Simha measures on degenerating families of Riemann surfaces.

In this paper, we propose a family of graph partition similarity measures that take the topology of the graph into account. These graph-aware measures are alternatives to using set partition similarity measures that are not specifically designed for graph partitions. The two types of measures, graph-aware and set parti…

Study measures rigidity for random walks and flows via generalized u-Gibbs states.

Stationary measures on hyperbolic surfaces with cusps are singular and stable under quasi-symmetries.

The study shows finite measure-preserving isometry groups for certain metric measure spaces.

The paper establishes a connection between different risk measures and their risk contributions.

Investigates set-valued risk measures for processes and vectors, proving equivalence and providing new dual representations.

Starting from the requirement that risk measures of financial portfolios should be based on their losses, not their gains, we define the notion of loss-based risk measure and study the properties of this class of risk measures. We characterize loss-based risk measures by a representation theorem and give examples of su…