On estimation of Hausdorff deviation of convex polygons in $\mathbb{R}^2$ from their differences with disks

We study a problem concerning the estimation of the Hausdorff deviation of convex polygons in $\mathbb R^2$ from their geometric difference with circles of sufficiently small radius. Problems with such a subject, in which not only convex polygons but also convex compacts in the Euclidean space $\mathbb R^n$ are considered, arise in various fields of mathematics and, in particular, in the theory of differential games, control theory, convex analysis. Estimates of Hausdorff deviations of convex compact sets in $\mathbb R^n$ in their geometric difference with closed balls in $\mathbb R^n$ are presented in the works of L.S. Pontryagin, his staff and colleagues. These estimates are very important in deriving an estimate for the mismatch of the alternating Pontryagin’s integral in linear differential games of pursuit and alternating sums. Similar estimates turn out to be useful in deriving an estimate for the mismatch of the attainability sets of nonlinear control systems in $\mathbb R^n$ and the sets approximating them. The paper considers a specific convex heptagon in $\mathbb R^2$. To study the geometry of this heptagon, we introduce the concept of a wedge in $\mathbb R^2$. On the basis of this notion, we obtain an upper bound for the Hausdorff deviation of a heptagon from its geometric difference with the disc in $\mathbb R^2$ of sufficiently small radius.

On one control problem with disturbance and vectograms depending linearly on given sets

A control problem with a given end time is considered, in which the control vectograms and disturbance depend linearly on the given convex compact sets. A multivalued mapping of the phase space of the control problem to the linear normed space E is given. The goal of constructing a control is that at the end of the control process the fixed vector of the space E belongs to the image of the multivalued mapping for any admissible realization of the disturbance. A stable bridge is defined in terms of multivalued functions. The presented procedure constructs, according to a given multivalued function which is a stable bridge, a control that solves the problem. Explicit formulas are obtained that determine a stable bridge in the considered control problem. Conditions are found under which the constructed stable bridge is maximal. Some problems of group pursuit can be reduced to the considered control problem with disturbance. The article provides such an example.

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.

Near-optimal recovery of linear and N-convex functions on unions of convex sets

In this paper we build provably near-optimal, in the minimax sense, estimates of linear forms and, more generally, ‘$N$-convex functionals’ (an example being the maximum of several fractional-linear functions) of unknown ‘signal’ from indirect noisy observations, the signal assumed to belong to the union of finitely many given convex compact sets. Our main assumption is that the observation scheme in question is good in the sense of Goldenshluger et al. (2015, Electron. J. Stat., 9, 1645–1712), the simplest example being the Gaussian scheme, where the observation is the sum of linear image of the signal and the standard Gaussian noise. The proposed estimates, same as upper bounds on their worst-case risks, stem from solutions to explicit convex optimization problems, making the estimates ‘computation-friendly’.

The completion of the space of convex, bounded sets with respect to the Demyanov metric

The Demyanov metric in the family of convex, compact sets in finite dimensional spaces has been recently extended to the family of convex, bounded sets – not necessarily closed. In this note it is shown that these spaces are not complete and a model for the completion is proposed. A full answer is given in ℝ