arXiv subjects

Dylan Langharst

Publications and source records attributed to Dylan Langharst.

24 records · Page 2Linked to original sources

arXiv2024

Weighted Berwald's Inequality

The inequality of Berwald is a reverse-Hölder like inequality for the $p$th average, $p\in (-1,\infty),$ of a non-negative, concave function over a convex body in $\mathbb{R}^n.$ We prove Berwald's inequality for averages of functions with respect to measures that have some concavity conditions, e.g. $s$-concave measures, $s\in \mathbb{R}.$ We also obtain equality conditions; in particular, this provides a new proof for the equality conditions of the classical inequality of Berwald. As applications, we generalize a number of classical bounds for the measure of the intersection of a convex body with a half-space and also the concept of radial means bodies and the projection body of a convex body.

math.MG↗

arXiv2024

The (Self-Similar, Variational) Rolling Stones

The interplay between variational functionals and the Brunn-Minkowski Theory is a well-established phenomenon widely investigated in the last thirty years. In this work, we prove the existence of solutions to the even logarithmic Minkowski problems arising from variational functionals, such as the first eigenvalue of the Laplacian and the torsional rigidity. In particular, we lay down a blueprint showing that the same result holds for more generic functionals by adapting the volume case from Böröczky, Lutwak, Yang, and Zhang. We show how these results imply the existence of self-similar solutions to variational flow problems à la Firey's worn stone problem.

math.FA↗

arXiv2023

Weighted Minkowski's Existence Theorem and Projection Bodies

The Brunn-Minkowski Theory has seen several generalizations over the past century. Many of the core ideas have been generalized to measures. With the goal of framing these generalizations as a weighted Brunn-Minkowski theory, we prove the Minkowski existence theorem for a large class of Borel measures with continuous density, denoted by $Λ^n$: for $ν$ a finite, even Borel measure on the unit sphere and even $μ\inΛ^n$, there exists a symmetric convex body $K$ such that $$dν(u)=c_{μ,K}dS^μ_{K}(u),$$ where $c_{μ,K}$ is a quantity that depends on $μ$ and $K$ and $dS^μ_{K}(u)$ is the surface area-measure of $K$ with respect to $μ$. Examples of measures in $Λ^n$ are homogeneous measures (with $c_{μ,K}=1$) and probability measures with radially decreasing densities (e.g. the Gaussian measure). We will also consider weighted projection bodies $Π_μK$ by classifying them and studying the isomorphic Shephard problem: if $μ$ and $ν$ are even, homogeneous measures with density and $K$ and $L$ are symmetric convex bodies such that $Π_μ K \subset Π_ν L$, then can one find an optimal quantity $\mathcal{A}>0$ such that $μ(K)\leq \mathcal{A}ν(L)$? Among other things, we show that, in the case where $μ=ν$ and $L$ is a projection body, $\mathcal{A}=1$.

math.FA↗

arXiv2023

Weighted Brunn-Minkowski Theory I: On Weighted Surface Area Measures

The Brunn-Minkowski theory in convex geometry concerns, among other things, the volumes, mixed volumes, and surface area measures of convex bodies. We study generalizations of these concepts to Borel measures with density in $\mathbb{R}^n$-- in particular, the weighted versions of mixed volumes (the so-called mixed measures) when dealing with up to three distinct convex bodies. We then formulate and analyze weighted versions of classical surface area measures, and obtain a new integral formula for the mixed measure of three bodies. As an application, we prove a Bézout-type inequality for rotational invariant log-concave measures, generalizing a result by Artstein-Avidan, Florentin and Ostrover. The results are new and interesting even for the special case of the standard Gaussian measure.

math.MG↗

arXiv2022

General Measure Extensions of Projection Bodies

The inequalities of Petty and Zhang are affine isoperimetric-type inequalities providing sharp bounds for $\text{vol}^{n-1}_{n}(K)\text{vol}_n(Π^\circ K),$ where $ΠK$ is a projection body of a convex body $K$. In this paper, we present a number of generalizations of Zhang's inequality to the setting of arbitrary measures. In addition, we introduce extensions of the projection body operator $Π$ to the setting of arbitrary measures and functions, while providing associated inequalities for this operator; in particular, Zhang-type inequalities. Throughout, we apply shown results to the standard Gaussian measure.

math.FA↗

arXiv2018

Probing context-dependent errors in quantum processors

Gates in error-prone quantum information processors are often modeled using sets of one- and two-qubit process matrices, the standard model of quantum errors. However, the results of quantum circuits on real processors often depend on additional external "context" variables. Such contexts may include the state of a spectator qubit, the time of data collection, or the temperature of control electronics. In this article we demonstrate a suite of simple, widely applicable, and statistically rigorous methods for detecting context dependence in quantum circuit experiments. They can be used on any data that comprise two or more "pools" of measurement results obtained by repeating the same set of quantum circuits in different contexts. These tools may be integrated seamlessly into standard quantum device characterization techniques, like randomized benchmarking or tomography. We experimentally demonstrate these methods by detecting and quantifying crosstalk and drift on the publicly accessible 16-qubit ibmqx3.

quant-ph↗