Related papers: Fixed points of Minkowski valuations

We consider the minimization problem for an integral functional $J$, possibly non-convex and non-coercive in $W^{1,1}_0(\Omega)$, where $\Omega\subset\R^n$ is a bounded smooth set. We prove sufficient conditions in order to guarantee that a…

Let $K$ be a smooth, origin-symmetric, strictly convex body in $\mathbb{R}^n$. If for some $\ell\in GL(n,\mathbb{R})$, the anisotropic Riemannian metric $\frac{1}{2}D^2 \Vert\cdot\Vert_{\ell K}^2$, encapsulating the curvature of $\ell K$,…

In this paper, we study first-order stationary monotone mean-field games (MFGs) with Dirichlet boundary conditions. While for Hamilton--Jacobi equations Dirichlet conditions may not be satisfied, here, we establish the existence of…

The Brunn-Minkowski inequality, applicable to bounded measurable sets $A$ and $B$ in $\mathbb{R}^d$, states that $|A+B|^{1/d} \geq |A|^{1/d}+|B|^{1/d}$. Equality is achieved if and only if $A$ and $B$ are convex and homothetic sets in…

Let $W(x,y) = a x^3 + b x^4 + f_5 x^5 + f_6 x^6 + (3 a x^2)^2 y + g_5 x^5 y + h_3 x^3 y^2 + h_4 x^4 y^2 + n_3 x^3 y^3 + a_{24} x^2 y^4 + a_{05} y^5 + a_{15} x y^5 + a_{06} y^6$, and $X=\frac{\partial W}{\partial x}$, $Y=\frac{\partial…

In Euclidean space, the generalised Minkowski problem asks, for a given finite Radon measure $\mu$ on the unit sphere $\mathbb{S}^d$, to find a compact convex set $K$ with area measure $\mu$. For convex sets in the Minkowski space invariant…

We study the fixed points of the Berezin transform on the Fock-type spaces $F_m^2$ with the weight $e^{-|z|^m}, m > 0.$ It is known that the Berezin transform is well-defined on the polynomials in $z$ and $\overline{z}$. In this paper we…

We consider the problem $F=f(\nu)$ for strictly convex, closed hypersurfaces in $S^{n+1}$ and solve it for curvature functions $F$ the inverses of which are of class $(K)$.

Central configurations play an important role in the dynamics of the $n$-body problem: they occur as relative equilibria and as asymptotic configurations in colliding trajectories. We illustrate how they can be found as projective fixed…

In this article, we derive a common fixed point result for a pair of single valued and set-valued mappings on a metric space having graphical structure. In this case, the set-valued map is assumed to be closed valued instead of closed and…

Realisations of associahedra with linearly non-isomorphic normal fans can be obtained by alteration of the right-hand sides of the facet-defining inequalities from a classical permutahedron. These polytopes can be expressed as Minkowski…

Consider a proper cocompact CAT(0) space X. We give a complete algebraic characterisation of amenable groups of isometries of X. For amenable discrete subgroups, an even narrower description is derived, implying Q-linearity in the…

We prove that there is a unique $p_0\in [0,1)$, which can be characterized by the eigenvalue of Hilbert operator related to a convex body, that the even $L^p$ Minkowski problem has a unique solution for $p\geq p_0$, and the uniqueness fails…

Given an action by a finite quantum group $\mathbb{G}$ on a von Neumann algebra $M$, we prove that a number of familiar $W^*$ properties are equivalent for $M$ and the fixed-point algebra $M^{\mathbb{G}}$ (i.e. hold or not simultaneously…

In this paper, we investigate the existence and uniqueness of fixed points for self-mappings defined on bipolar metric spaces using a new class of contractive conditions, namely polynomial-type contractions. Our main results establish…

We consider irrational fixed points of the Minkowski question mark function $? (x)$, that is irrational solutions of the equation $? (x)=x$. It is easy to see that there exist at least two such points. Although it is not known if there are…

This article investigates sharp comparison of moments for various classes of random variables appearing in a geometric context. In the first part of our work we find the optimal constants in the Khintchine inequality for random vectors…

We investigate the relevance of the conformal method by investigating stability issues for the Einstein-Lichnerowicz conformal constraint system in a nonlinear scalar-field setting. We prove the stability of the system with respect to…

We propose a method for the treatment of two--point boundary value problems given by nonlinear ordinary differential equations. The approach leads to sequences of roots of Hankel determinants that converge rapidly towards the unknown…

In this paper we present some fixed-figure theorems as a geometric approach to the fixed-point theory when the number of fixed points of a self-mapping is more than one. To do this, we modify the Jleli-Samet type contraction and define new…