Russian Mathematical Surveys, 2024, Volume 79, Issue 2, Pages 361–362 DOI: https://doi.org/10.4213/rm10167e (Mi rm10167)

DOI: https://doi.org/10.4213/rm10167e

Funding agency	Grant number
Russian Science Foundation	22-11-00129
This research was carried out at Lomonosov Moscow State University with the financial support of the Russian Science Foundation (grant no. 22-11-00129).

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A set is a Chebyshev set if it is a set of existence and uniqueness, that is, each point has a unique best approximant in this set. We study properties of Chebyshev sets composed of finitely or countably many planes (a plane is a closed affine subspace possibly degenerate to a point). In parallel with usual normed linear spaces, we also consider spaces with asymmetric norm. An asymmetric norm $\|\cdot|$ on a real linear space $X$ is defined by the following axioms: (1) $\|\alpha x|=\alpha\| x|$ for all $\alpha\geqslant 0$, $x\in X$; (2) $\|x+y|\leqslant \|x|+\|y|$ for all $x,y\in X$; (3) $\|x|\geqslant 0$ for all $x\in X$, $\|x|=0\Leftrightarrow x=0$.

We assume that the union of planes $\bigcup_{i\in A} L_i$ is irreducible, that is, for each $j\in A$, $\operatorname{card}A\geqslant 2$, the set $L_j\setminus \bigcup_{i\in A, \, i\ne j} L_i$ is dense in $L_j$ with respect to the symmetrization norm $\|x\|_{\rm sym}=\max\{\|x|,\|-x|\}$. By Baire’s category theorem, each at most countable union of planes $\bigcup_iL_i$, $L_i\not\subset L_j$, $i\ne j$, in a Banach space is irreducible. If a union of planes $\bigcup_{i\in A} L_i$ is irreducible, then no plane in this union contains any other plane.

Irreducible unions of planes are being actively studied and have numerous applications, for example, to signal recovery problems (in many cases signals lie on a union of planes), the recovery of signals from sparse baseband signals, as well as to problems in mathematical economics and related problems of rank minimization and $\ell_0$-norm minimization. Unions of planes also appear in ridge-function approximation and in classical variants of $n$-term approximation.

Tsar’kov ([4], § 3) studied Chebyshev sets composed of at most countably many approximatively compact sets. In particular, he showed that if $X$ is a uniformly smooth Banach space, and $M\subset X$ is a Chebyshev set composed of at most countably many approximatively sets, then $M$ is a Chebyshev sun. If, in addition, $X$ is smooth, then $M$ is convex. Our results continue and generalize the investigations on approximation by unions of planes begun in [3] and [4]. We follow the definitions from [1]–[3].

A set $M\ne\varnothing$ is unimodal (or an $\mathrm{LG}$-set) if for each $x\notin M$ each local minimum of the function $\|y-x|$, $y\in M$, is global. A set is $B$-connected if its intersection with any closed ball is connected. A set $M$ is $\mathring{B}$-complete if for all $x\in X$ and $r>0$ the condition $M_0:=(\mathring{B}(x,r)\cap M) \ne\varnothing$ implies that $\overline{M}_0\supset (M\cap B(x,r))$ (here $\mathring{B}(x,r)=\{y\in X \mid \|y-x| < r\}$ is an open ball and $B(x,r)=\{y\in X \mid \|y-x|\leqslant r\}$).

A symmetrizable asymmetric space is a CLUR-space (written $X\in (\mathrm{CLUR})$) if for $x\in S:=\{s\mid \|s\,|=1\}$ and $y_n\in S$ the convergence $\|x+y_n\,|\to 2$ implies that $(y_n)$ has a convergent subsequence. A reflexive CLUR-space is a Efimov–Stechkin space. The class (CLUR) includes the uniformly normed convex spaces and the locally uniformly convex normed spaces (in particular, the Hilbert spaces and the $L^p$-spaces, $1<p<\infty$).

The next result gives a partial answer to the well-known Efimov–Stechkin–Klee problem on the convexity of Chebyshev sets composed of finitely or countably many planes.

For Chebyshev sets composed of finitely many planes an answer to this problem is given in the most general case without any restriction on the space or the set. In addition, the space is not assumed to be complete.

The following application of the above results is worth pointing out: in $L^1(\Omega,\sigma,\mu)$ with atomless $\sigma$-finite measure $\mu$, an at most countable irreducible union of planes of finite dimension or codimension is never a Chebyshev set.

Bibliography
1.	A. R. Alimov, K. S. Ryutin, and I. G. Tsar'kov, Russian Math. Surveys, 78:3 (2023), 399–442
2.	A. R. Alimov and I. G. Tsarkov, Filomat, 38:9 (2024), 3243–3259
3.	A. R. Alimov and I. G. Tsar'kov, J. Approx. Theory, 298 (2024), 106009, 12 pp.
4.	I. G. Tsar'kov, Math. Notes, 113:6 (2023), 840–849

Citation in format AMSBIB

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\by A.~R.~Alimov, I.~G.~Tsar'kov
\paper Chebyshev sets that are unions of planes
\jour Russian Math. Surveys
\yr 2024
\vol 79
\issue 2
\pages 361--362
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