Invariant Convex Sets Research Articles

Convex Stone-Weierstrass theorems and invariant convex sets

A \textit {convex polynomial} is a convex combination of the monomials $\{1, x, x^2, \ldots \}$. This paper establishes that the convex polynomials on $\mathbb R$ are dense in $L^p(\mu )$ and weak$^*$ dense in $L^\infty (\mu )$ whenever $\mu $ is a compactly supported regular Borel measure on $\mathbb {R}$ and $\mu ([-1,\infty )) = 0$. It is also shown that the convex polynomials are norm dense in $C(K)$ precisely when $K \cap [-1, \infty ) = \emptyset $, where $K$ is a compact subset of the real line. Moreover, the closure of the convex polynomials on $[-1,b]$ is shown to be the functions that have a convex power series representation. A continuous linear operator $T$ on a locally convex space $X$ is \textit {convex-cyclic} if there is a vector $x \in X$ such that the convex hull of the orbit of $x$ is dense in $X$. The previous results are used to characterize which multiplication operators on various real Banach spaces are convex-cyclic. Also, it is shown for certain multiplication operators that every nonempty closed invariant convex set is a closed invariant subspace.

Convex-cyclic matrices, convex-polynomial interpolation and invariant convex sets

We define a convex-polynomial to be one that is a convex combination of the monomials $\{1, z, z^2, \ldots\}$. This paper explores the intimate connection between peaking convex-polynomials, interpolating convex-polynomials, invariant convex sets, and the dynamics of matrices. In particular, we use these intertwined relations to both prove which matrices are convex-cyclic while at the same time proving that we can prescribe the values and a finite number of the derivatives of a convex-polynomial subject to certain natural constraints. These properties are also equivalent to determining those matrices whose invariant closed convex sets are all invariant subspaces. Our characterization of the convex-cyclic matrices gives a new and correct proof of a similar result by Rezaei that was stated and proven incorrectly.

Convex-cyclic matrices, convex-polynomial interpolation and invariant convex sets

Convex-cyclic matrices, convex-polynomial interpolation and invariant convex sets

The equivariant Minkowski problem in Minkowski space

The classical Minkowski problem in Minkowski space asks, for a positive function $\phi$ on $\mathbb{H}^d$, for a convex set $K$ in Minkowski space with $C^2$ space-like boundary $S$, such that $\phi(\eta)^{-1}$ is the Gauss--Kronecker curvature at the point with normal $\eta$. Analogously to the Euclidean case, it is possible to formulate a weak version of this problem: given a Radon measure $\mu$ on $\mathbb{H}^d$ the generalized Minkowski problem in Minkowski space asks for a convex subset $K$ such that the area measure of $K$ is $\mu$. In the present paper we look at an equivariant version of the problem: given a uniform lattice $\Gamma$ of isometries of $\mathbb{H}^d$, given a $\Gamma$ invariant Radon measure $\mu$, given a isometry group $\Gamma_{\tau}$ of Minkowski space, with $\Gamma$ as linear part, there exists a unique convex set with area measure $\mu$, invariant under the action of $\Gamma_{\tau}$. The proof uses a functional which is the covolume associated to every invariant convex set. This result translates as a solution of the Minkowski problem in flat space times with compact hyperbolic Cauchy surface. The uniqueness part, as well as regularity results, follow from properties of the Monge--Amp\`ere equation. The existence part can be translated as an existence result for Monge--Amp\`ere equation. The regular version was proved by T.~Barbot, F.~B\'eguin and A.~Zeghib for $d=2$ and by V.~Oliker and U.~Simon for $\Gamma_{\tau}=\Gamma$. Our method is totally different. Moreover, we show that those cases are very specific: in general, there is no smooth $\Gamma_\tau$-invariant surface of constant Gauss-Kronecker curvature equal to $1$.

Traveling wave solutions in a diffusive producer-scrounger model

This paper looks into the stability of equilibria, existence and non-existence of traveling wave solutions in a diffusive producer-scrounger model. We find that the existence and non-existence of traveling wave solutions are determined by a minimum wave speed \begin{document}$c_{m}$\end{document} and a threshold value \begin{document}$R_{0}$\end{document} . By constructing a suitable invariant convex set $Γ$ and applying Schauder fixed point theorem, the existence for \begin{document}$c>c_{m}, R_{0}>1$\end{document} was established. Besides, a Lyapunov function is constructed subtly to explore the asymptotic behaviors of traveling wave solutions. The non-existences of traveling wave solutions for both \begin{document}$c 1$\end{document} and \begin{document}$R_{0}≤1, c > 0$\end{document} were obtained by two-sides Laplace transform and reduction method to absurdity.

Invariant convex sets in polar representations

We study a compact invariant convex set E in a polar representation of a compact Lie group. Polar representations are given by the adjoint action of K on p, where K is a maximal compact subgroup of a real semisimple Lie group G with Lie algebra g = k ⊕ p. If a ⊂ p is a maximal abelian subalgebra, then P = E ∩ a is a convex set in a. We prove that up to conjugacy the face structure of E is completely determined by that of P and that a face of E is exposed if and only if the corresponding face of P is exposed. We apply these results to the convex hull of the image of a restricted1 momentum map.

Coxeter group in Hilbert geometry

A theorem of Tits and Vinberg allows to build an action of a Coxeter group € \Gamma on a properly convex open set \Omega of the real projective space, thanks to the data P of a polytope and reflection across its facets. We give sufficient conditions for such action to be of finite covolume, convex-cocompact or geometrically finite. We describe a hypothesis that makes those conditions necessary. Under this hypothesis, we describe the Zariski closure of € \Gamma , nd the maximal € \Gamma -invariant convex set, when there is a unique € \Gamma -invariant convex set, when the convex set \Omega  is strictly convex, when we can find a € \Gamma -invariant convex set  \Omega ' which is strictly convex.

Research Spotlights

From social networks to machine learning to biological network analysis, graphs spring up in all sorts of applications related to either the natural sciences or systems created by humans. While individual nodes and edges express relationships between interacting entities, there is often far greater insight in understanding a graph's larger structural properties, e.g., treewidths, the maximal degree, or the second smallest eigenvalue of an associated graph Laplacian. As these fundamental structural properties are independent of a specific node labeling, it is then worthwhile to examine graph invariants, functions of graphs that do not depend on specific node labels. The paper by Chandrasekaran, Parrilo, and Willsky explores the realm of convex graph invariants, convex functions of a graph's adjacency matrix which are also graph invariants. These convex graph invariants arise in many practical situations and are advantageous in that one can leverage convex optimization ideas and tools to provide tractable algorithms. Three canonical graph structural problems are introduced: graph deconvolution, graph generation, and graph hypothesis testing. Deconvolution seeks to unravel a graph into simpler graph structures. Network analysis in areas such as biological sciences and social networks often rely on this type of decomposition. Graph generation seeks to produce graphs satisfying specified constraints (e.g., each node has a certain degree or the corresponding Laplacian matrix has certain spectral properties). This might arise in situations where one wishes to construct an economical network with certain robustness characteristics. Graph hypothesis testing aims to categorize candidate graphs, i.e., deciding whether they fit better within one graph family or another (where families are specified by graph invariants). The paper argues that the key to understanding and computing convex graph invariants lies in being able to first represent them via elementary invariants. This leads to solution methodologies associated with quadratic assignment problems and tractable approximations via optimization techniques such as linear programming and semidefinite programming. Overall, the paper provides a clear view into the world of convex graph invariants. In addition to providing a general framework for computing these invariants, numerous examples introduce readers unfamiliar with this area to the wide range of practical cases where the notions of invariant convex sets and convex graph invariants emerge.

Rigidity of invariant convex sets in symmetric spaces

The main result implies that a proper convex subset of an irreducible higher rank symmetric space cannot have Zariski dense stabilizer.

Invariant convex sets and functions in Lie algebras

We say that an invariant convex coneW in a Lie algebras\(\mathfrak{g}\) is elliptic if its interior consists of elliptic elements of\(\mathfrak{g}\). If such a cone exists, then\(\mathfrak{g}\) has a compactly embedded Cartan subalgebra. The first main result, of this paper is a characterization of those Lie algebras, which contain elliptic invariant cones. If\(D \subseteq W\) is an invariant domain in such a cone, then we characterize the invariant locally convex functions onD by their restrictions to\(D \cap \mathfrak{t}\) where\(\mathfrak{t}\) is a compactly embedded Cartan subalgebra.

Inequalities for probability contents of convex sets via geometric average

It is shown that: If ( X 1, X 2) is a permutation invariant central convex unimodal random vector and if A is a symmetric (about 0) permutation invariant convex set then P{(aX 1, X 2 a ) ∈ A} is nondecreasing as a varies from )+ to 1 and is non-increasing as a varies from 1 to ∞ (that is, P{( a 1 X 1, a 2 X 2) ϵ A} is a Schur-concave function of (log a 1, log a 2). Some extensions of this result for the n-dimensional case are discussed. Applications are given for elliptically contoured distributions and scale parameter families.

The Integral of an Invariant Unimodal Function over an Invariant Convex Set--An Inequality and Applications

The Integral of an Invariant Unimodal Function over an Invariant Convex Set--An Inequality and Applications

The integral of an invariant unimodal function over an invariant convex set—an inequality and applications

The integral of an invariant unimodal function over an invariant convex set—an inequality and applications