Related papers: On a Minimax Problem for Ovals

We derive $H_{\text{curl}}$-error estimates and improved $L^2$-error estimates for the Maxwell equations approximated using edge finite elements. These estimates only invoke the expected regularity pickup of the exact solution in the scale…

For a triangle $\Delta$, let (P) denote the problem of the existence of points in the plane of $\Delta$, that are at rational distance to the 3 vertices of $\Delta$. Answer to (P) is known to be positive in the following situation: $\Delta$…

In this paper we investigate the approximation of a diffusion model problem with contrasted diffusivity and the error analysis of various nonconforming approximation methods. The essential difficulty is that the Sobolev smoothness index of…

Let $\Omega$ be a pseudoconvex domain in $\mathbb C^n$ satisfying an $f$-property for some function $f$. We show that the Bergman metric associated to $\Omega$ has the lower bound $\tilde g(\delta_\Omega(z)^{-1})$ where $\delta_\Omega(z)$…

We prove some sharp isoperimetric type inequalities for domains with smooth boundary on Riemannian manifolds. For example, using generalized convexity, we show that among all domains with a lower bound $l$ for the cut distance and Ricci…

We prove that the results regarding the Isoperimetric inequality and Cheeger constant formulated in terms of the Minkowski content, obtained by the authors in previous papers in the framework of essentially non-branching metric measure…

We prove that the isoperimetric profile of a convex domain $\Omega$ with compact closure in a Riemannian manifold $(M^{n+1},g)$ satisfies a second order differential inequality which only depends on the dimension of the manifold and on a…

This paper studies sets of matrices induced by quadratic inequalities. In particular, the center and radius of a smallest ball containing the set, called a Chebyshev center and the Chebyshev radius, are studied. In addition, this work…

The classical isoperimetric inequality in the Euclidean plane $\mathbb{R}^2$ states that for a simple closed curve $M$ of the length $L_{M}$, enclosing a region of the area $A_{M}$, one gets \begin{align*} L_{M}^2\geqslant 4\pi A_{M}.…

Both unconstrained and constrained minimax single facility location problems are considered in multidimensional space with Chebyshev distance. A new solution approach is proposed within the framework of idempotent algebra to reduce the…

Minimizing a convex, quadratic objective of the form $f_{\mathbf{A},\mathbf{b}}(x) := \frac{1}{2}x^\top \mathbf{A} x - \langle \mathbf{b}, x \rangle$ for $\mathbf{A} \succ 0 $ is a fundamental problem in machine learning and optimization.…

In this paper we consider a partial overdetermined mixed boundary value problem in domains inside a cone as in [18]. We show that in cones having an isoperimetric property the only domains which admit a solution and which minimize a…

In this paper we establish a general form of the isoperimetric inequality for immersed closed curves (possibly non-convex) in the plane under rotational symmetry. As an application we obtain a global existence result for the surface…

We combine geometric methods with numerical box search algorithm to show that the minimal area of a convex set on the plane which can cover every closed plane curve of unit length is at least 0.0975. This improves the best previous lower…

Let $n,m\ge 1$, $\alpha\in(0,1)$, and $\beta\ge 0$. For the Grushin-type operator \[ L=-\nabla_x\!\cdot\!\bigl(|x|^{2\alpha}\nabla_x\bigr)+|x|^{2\beta}\Delta_y \qquad \text{on } \mathbb R^n\times \mathbb R^m, \] we prove the isoperimetric…

We consider the problem of domain approximation in finite element methods for Maxwell equations on curved domains, i.e., when affine or polynomial meshes fail to exactly cover the domain of interest. In such cases, one is forced to…

We consider the punctured plane with volume density $|x|^\alpha$ and perimeter density $|x|^\beta$. We show that centred balls are uniquely isoperimetric for indices $(\alpha,\beta)$ which satisfy the conditions $\alpha-\beta+1>0$,…

We define a linear code $C_\eta(\delta_T,\delta_X)$ by evaluating polynomials of bidegree $(\delta_T,\delta_X)$ in the Cox ring on $\mathbb{F}_q$-rational points of the Hirzebruch surface of parameter $\eta$ on the finite field…

Benguria and Loss have conjectured that, amongst all smooth closed curves of length $2\pi$ in the plane, the lowest possible eigenvalue of the operator $L=-\Delta+\kappa^2$ was one. They observed that this value was achieved on a…

Consider an operator that takes the Fourier transform of a discrete measure supported in $\mathcal{X}\subset[-\frac 12,\frac 12)^d$ and restricts it to a compact $\Omega\subset\mathbb{R}^d$. We provide lower bounds for its smallest singular…