Compact Convex Set Research Articles

Оценка времени преследования в дифференциальных играх многих игроков на выпуклом компакте

Статья посвящена изучению задачи построения стратегии преследования в простых дифференциальных играх многих лиц с фазовыми ограничениями в состоянии игроков, в смысле попадания в некоторый окрест беглеца. Игра происходит в n-мерном евклидовом пространстве на выпуклом компакте. Рассматривается задача преследования, когда число преследователей n–1, то есть меньше n, в смысле — l-поимка. Предложена структура построения управлений преследования, которая обеспечит завершение игры за конечное время. Получена оценка сверху времени игры для завершения преследования. Рассмотрена вспомогательная задача простого преследования на единичном кубе в первом ортанте и построена стратегия преследователей для завершения игры с особыми начальными позициями. Полученные результаты применяются для решения дифференциальных игр с произвольными начальными позициями. Для этой задачи предложена структура построения стратегии преследования, которая обеспечит завершение игры за конечное время. Также рассматривается обобщение задачи в смысле усложнения помехи. Рассматривается более общая задача простого преследования на кубе произвольного размера в первом ортанте. С помощью предложенных стратегий доказаны возможности завершения преследования и получена оценка времени. Как следствие этого результата, получены оценки снизу и сверху для времени преследования в игре с препятствиями типа шара. Получены оценки времени преследования, когда компакт – произвольно выпуклое множество. Определено понятие выпуклого множества относительно направления сечения, которое необязательно выпукло. В нем изучена задача простого преследования в дифференциальной игре многих игроков и показаны возможности завершения преследования с применением предложенной стратегии. Оценивается сверху время завершения преследования данной игры.

Existence of nice resolutions in Cp(X) and its bidual often implies metrizability of Cp(X)

Let X be a Tychonoff space. We show that Cp(X), the space of real-valued continuous functions on X equipped with the pointwise topology, admits a resolution (an ordered covering indexed by NN) consisting of convex compact sets that swallows the local null sequences if and only if X is countable and discrete. Then we prove (main result) (i) the weak* bidual of Cp(X) admits a resolution consisting of bounded sets if and only if X is countable. Hence (ii) if Cp(X) admits a fundamental bounded resolution, X must be countable [8]. If X is first countable, then Cp(X) admits a resolution made up of bounded sets swallowing the Cauchy sequences if and only if X is countable. In the context of the present research, the weak* bidual of Cp(X) has received little or null attention so far. Result (i) fixes this situation.

On M-predecomposable sets

M-predecomposable set in $${\mathbf {R}}^n$$ is a convex set which is the sum of a compact convex set and a convex (not necessarily closed) cone. Complementing the existing results on M-predecomposable sets, we study their support and asymptotic properties. Also, we establish some properties of M-polyhedral sets, which are sums of compact convex sets and polyhedral cones.

Constructing a subgradient from directional derivatives for functions of two variables

For any scalar-valued bivariate function that is locally Lipschitz continuous and directionally differentiable, it is shown that a subgradient may always be constructed from the function's directional derivatives in the four compass directions, arranged in a so-called "compass difference". When the original function is nonconvex, the obtained subgradient is an element of Clarke's generalized gradient, but the result appears to be novel even for convex functions. The function is not required to be represented in any particular form, and no further assumptions are required, though the result is strengthened when the function is additionally L-smooth in the sense of Nesterov. For certain optimal-value functions and certain parametric solutions of differential equation systems, these new results appear to provide the only known way to compute a subgradient. These results also imply that centered finite differences will converge to a subgradient for bivariate nonsmooth functions. As a dual result, we find that any compact convex set in two dimensions contains the midpoint of its interval hull. Examples are included for illustration, and it is demonstrated that these results do not extend directly to functions of more than two variables or sets in higher dimensions.

Hadamard directional differentiability of the optimal value of a linear second-order conic programming problem

<p style='text-indent:20px;'>In this paper, we consider perturbation properties of a linear second-order conic optimization problem and its Lagrange dual in which all parameters in the problem are perturbed. We prove the upper semi-continuity of solution mappings for the pertured problem and its Lagrange dual problem. We demonstrate that the optimal value function can be expressed as a min-max optimization problem over two compact convex sets, and it is proven as a Lipschitz continuous function and Hadamard directionally differentiable.</p>

Best proximity version of Krasnoselskii’s fixed point theorem

We prove a best proximity point version of Krasnoselskii’s fixed point theorem. As a consequence we obtain the existence of best proximity points of relatively u-continuous maps for a pair of compact convex sets.

(In)homogeneous invariant compact convex sets of probability measures

It is proved that for any iterated function system of contractions on a complete metric space there exists an invariant compact convex sets of probability measures of compact support on this space. A similar result is proved for the inhomogeneous compact convex sets of probability measures of compact support.

Periodic Time-Optimal Controls on Two-Step Free-Nilpotent Lie Groups

For two-step free nilpotent Lie algebras, we describe symplectic foliations and Casimir functions. A left-invariant time-optimal problem is considered in which the set of admissible controls is given by a strictly convex compact set in the first layer of the Lie algebra that contains the origin in its interior. We describe integrals for the vertical subsystem of the Hamiltonian system of the Pontryagin maximum principle. The properties of solutions to this system for low ranks of the Poisson bivector are described.

A Comprehensive Review of Fixed Point Theory: Foundations, Applications, and Emerging Trends in Mathematical Spaces

Fixed point theory is a crucial branch of mathematical analysis that investigates the conditions under which a function returns a point to itself, symbolizing stability and equilibrium. This paper delves into various fixed point theorems within the contexts of metric spaces, Banach spaces, and Hilbert spaces, emphasizing their foundational importance and wide-ranging applications. The Banach Fixed Point Theorem guarantees the existence and uniqueness of fixed points for contraction mappings in complete metric spaces, while Brouwer's Fixed Point Theorem states that any continuous function mapping a compact convex set to it will have at least one fixed point. Recent developments have expanded these classical results to encompass new types of contraction mappings and generalized distance functions, enhancing their relevance in dynamic systems, control theory, and optimization challenges. Additionally, the paper discusses the Schauder Fixed Point Theorem in Banach spaces, highlighting its significance in analysing nonlinear operators. In Hilbert spaces, fixed point results are examined in relation to nonlinear integral equations and optimization methods, showcasing their practical implications in engineering and variation techniques. Emerging trends include the study of fixed point results in fuzzy and probabilistic environments, as well as the integration of computational approaches with traditional fixed point methods. This paper illustrates the continuous evolution of fixed point theory, connecting abstract mathematical principles with practical problem-solving across various fields. Finally, it proposes future research directions to further explore the potential of fixed point theory in modern mathematics.

Furstenberg boundaries for pairs of groups

Furstenberg has associated to every topological group $G$ a universal boundary $\unicode[STIX]{x2202}(G)$. If we consider in addition a subgroup $H<G$, the relative notion of $(G,H)$-boundaries admits again a maximal object $\unicode[STIX]{x2202}(G,H)$. In the case of discrete groups, an equivalent notion was introduced by Bearden and Kalantar (Topological boundaries of unitary representations. Preprint, 2019, arXiv:1901.10937v1) as a very special instance of their constructions. However, the analogous universality does not always hold, even for discrete groups. On the other hand, it does hold in the affine reformulation in terms of convex compact sets, which admits a universal simplex $\unicode[STIX]{x1D6E5}(G,H)$, namely the simplex of measures on $\unicode[STIX]{x2202}(G,H)$. We determine the boundary $\unicode[STIX]{x2202}(G,H)$ in a number of cases, highlighting properties that might appear unexpected.

An Inverse Mapping Theorem in Fréchet-Montel Spaces

Influenced by a recent note by M. Ivanov and N. Zlateva, we prove a statement in the style of Nash-Moser-Ekeland theorem for mappings from a Frechet-Montel space with values in any Frechet space (not necessarily standard). The mapping under consideration is supposed to be continuous and directionally differentiable (in particular Gateaux differentiable) with the derivative having a right inverse. We also consider an approximation by a graphical derivative and by a linear operator in the spirit of Graves’ theorem. Finally, we derive corollaries of the abstract results in finite dimensions. We obtain, in particular, sufficient conditions for the directional semiregularity of a mapping defined on a (locally) convex compact set in directions from a locally conic set; and also conditions guaranteeing that the nonlinear image of a convex set contains a prescribed ordered interval.

Compatibility of any pair of 2-outcome measurements characterizes the Choquet simplex

For a compact convex subset $K $ of a locally convex Hausdorff space, a measurement on $A(K)$ is a finite family of positive elements in $A(K)$ normalized to the unit constant $1_K$, where $A(K)$ denotes the set of continuous real affine functionals on $K$. It is proved that a compact convex set $K$ is a Choquet simplex if and only if any pair of $2$-outcome measurements are compatible, i.e.\ the measurements are given as the marginals of a single measurement. This generalizes the finite-dimensional result of [Pl\'avala M 2016 Phys.\ Rev.\ A \textbf{94}, 042108] obtained in the context of the foundations of quantum theory.

Asymptotic properties of the (convex) hyperspaces

It is known that the hyperspaces of compact sets and compact convex set of the Euclidean space $\mathbb R^n$, $n\ge2$, both are homeomorphic to the puctured Hilbert cube.
 The main result of this note states that these hyperspaces are not coarsely equivalent.

Nondifferentiable higher-order duality theorems for new type of dual model under generalized functions

The motivation behind this article is to study a class of nondifferentiable multiobjective fractional programming problem in which each component of objective functions contains a term including the support function of a compact convex set. For a differentiable function, we consider a class of higher order pseudo quasi/ strictly pseudo quasi/weak strictly pseudo quasi- (V, ρ, d)-type-I convex functions. Under these the higher-order pseudo quasi/ strictly pseudo quasi/weak strictly pseudo quasi- (V, ρ, d)-type-I convexity assumptions, we prove the higher-order weak, higher-order strong and higher-order converse duality theorems related to efficient solution.

On constrained Volterra cubic stochastic operators

We consider constrained Volterra cubic stochastic operators and construct several Lyapunov functions for the constrained Volterra cubic stochastic operators. We prove that such kind operators do not have periodic trajectories. Finally, we show that the set of all constrained Volterra cubic stochastic operators is a convex compact set and find the extreme points of this set.

Uniqueness of the measurement function in Crofton's formula

Crofton's intersection formula states that the (n−j)th intrinsic volume of a compact convex set in Rn can be obtained as an invariant integral of the (k−j)th intrinsic volume of sections with k-planes. This paper discusses the question if the (k−j)th intrinsic volume can be replaced by other functionals, that is, if the measurement function in Crofton's formula is unique.The answer is negative: we show that the sums of the (k−j)th intrinsic volume and certain translation invariant continuous valuations of homogeneity degree k yield counterexamples. If the measurement function is local, these turn out to be the only examples when k=1 or when k=2 and we restrict considerations to even measurement functions. Additional examples of local functionals can be constructed when k≥2.

Construction of labyrinths in pseudoconvex domains

We build in a given pseudoconvex (Runge) domain D of $${\mathbb {C}}^N$$ an $$\mathcal O(D)$$ -convex set $$\Gamma $$ , every connected component of which is a holomorphically contractible (convex) compact set, enjoying the property that any continuous path $$\gamma :[0,1)\rightarrow D$$ with $$\lim _{r\rightarrow 1}\gamma (r)\in \partial D$$ and omitting $$\Gamma $$ has infinite length. This solves a problem left open in a recent paper by Alarcon and Forstneric.

On M-decomposable sets

According to Goberna, González, Martínez-Legaz, and Todorov (2010), an M-decomposable set in Rn is a closed convex set which is the sum of a compact convex set and a closed convex cone. Complementing the existing results on M-decomposable sets, we study their extreme, exposed, and asymptotic properties. Also, we consider M-polyhedral sets which are sums of compact convex sets and polyhedral cones and establish some characteristic properties of such sets.

Metric sub-regularity in optimal control of affine problems with free end state

The paper investigates the property of Strong Metric sub-Regularity (SMsR) of the mapping representing the first order optimality system for a Lagrange-type optimal control problem which is affine with respect to the control. The terminal time is fixed, the terminal state is free, and the control values are restricted in a convex compact set U. The SMsR property is associated with a reference solution of the optimality system and ensures that small additive perturbations in the system result in solutions which are at distance to the reference one, at most proportional to the size of the perturbations. A general sufficient condition for SMsR is obtained for appropriate space settings and then specialized in the case of a polyhedral set U and purely bang-bang reference control. Sufficient second-order optimality conditions are obtained as a by-product of the analysis. Finally, the obtained results are utilized for error analysis of the Euler discretization scheme applied to affine problems.

Traversable Wormholes and the Brouwer Fixed-Point Theorem

The Brouwer fixed-point theorem in topology states that for any continuous mapping $f$ on a compact convex set into itself admits a fixed point, i.e., a point $x_0$ such that $f(x_0)=x_0$. Under certain conditions, this fixed point corresponds to the throat of a traversable wormhole, i.e., $b(r_0)=r_0$ for the shape function $b=b(r)$. The possible existence of wormholes can therefore be deduced from purely mathematical considerations without going beyond the existing physical requirements.