Related papers: Continuous Convexity Measures
Contraction metrics are crucial in control theory because they provide a powerful framework for analyzing stability, robustness, and convergence of various dynamical systems. However, identifying these metrics for complex nonlinear systems…
In the first part of this paper, we develop the theory of anisotropic curvature measures for convex bodies in the Euclidean space. It is proved that any convex body whose boundary anisotropic curvature measure equals a linear combination of…
The paper is devoted to the study of Gromov-Hausdorff convergence and stability of irreversible metric-measure spaces, both in the compact and noncompact cases. While the compact setting is mostly similar to the reversible case developed by…
The subgradient method is one of the most fundamental algorithmic schemes for nonsmooth optimization. The existing complexity and convergence results for this method are mainly derived for Lipschitz continuous objective functions. In this…
In this paper, we revisit the notion of length measures associated to planar closed curves. These are a special case of area measures of hypersurfaces which were introduced early on in the field of convex geometry. The length measure of a…
Diversities have been recently introduced as a generalization of metrics for which a rich tight span theory could be stated. In this work we take up a number of questions about hyperconvexity, diversities and fixed points of nonexpansive…
The purpose of this note is to record a consequence, for general metric spaces, of a recent result of David Bate. We prove the following fact: Let $X$ be a compact metric space of topological dimension $n$. Suppose that the $n$-dimensional…
Estimation of convex functions finds broad applications in engineering and science, while convex shape constraint gives rise to numerous challenges in asymptotic performance analysis. This paper is devoted to minimax optimal estimation of…
This paper builds upon our prior formalisation of R^n in ACL2(r) by presenting a set of theorems for reasoning about convex functions. This is a demonstration of the higher-dimensional analytical reasoning possible in our metric space…
We develop a rigorous framework for global non-convex optimization by reformulating the minimization problem as a discounted infinite-horizon optimal control problem. For non-convex, continuous, and possibly non-smooth objective functions…
We present mathematical methods, based on convex optimization, for correcting non-physical coherency matrices measured in polarimetry. We also develop the method for recovering the coherency matrices corresponding to the smallest and…
The growing complexity of dynamical systems and advances in data collection necessitates robust data-driven control strategies without explicit system identification and robust synthesis. Data-driven stability has been explored in linear…
The classical theory of regularity of embeddings of compact convex sets was developed in the 1970s, exclusively in the real case, and even there it does not appear to have been stated in its simplest form. We begin by revisiting this…
Cantor sets in \(\mathbb{R}\) are common examples of sets for which Hausdorff measures can be positive and finite. However, there exist Cantor sets for which no Hausdorff measure is supported and finite. The purpose of this paper is to try…
We introduce a compactification of the space of simple positive divisors on a Riemann surface, as well as a compactification of the universal family of punctured surfaces above this space. These are real manifolds with corners. We then…
Recent advances in convex optimization have led to new strides in the phase retrieval problem over finite-dimensional vector spaces. However, certain fundamental questions remain: What sorts of measurement vectors uniquely determine every…
Some concepts, such as non-compactness measure and condensing operators, defined on metric spaces are extended to uniform spaces. Such extensions allow us to locate, in the context of uniform spaces, some classical results existing in…
Lossless Convexification (LCvx) is a modeling approach that transforms a class of nonconvex optimal control problems, where nonconvexity primarily arises from control constraints, into convex problems through convex relaxations. These…
On a compact three-dimensional Riemannian manifold with boundary, we prove the compactness of the full set of conformal metrics with positive constant scalar curvature and constant mean curvature on the boundary. This involves a blow-up…
There is a hierarchy of structure conditions for convex sets. In this paper we study a recently defined [3, 8, 9] condition called locally nonconical convexity (abbreviated LNC). Is is easy to show that every strictly convex set is LNC, as…