Minimizers for Banach-Space Subspace Segmentation

Definitions and mathematical statement

Let $X$ be a Banach space and let $M$ be a positive integer. Write \[ \mathcal S_M(X)=\{(V_1,\ldots,V_M): V_j\subseteq X\text{ is a closed linear subspace for each }j\}. \] The source asks for necessary and sufficient conditions for the subspace-segmentation approximation error with exponent $p\ne2$ to attain its minimum over such tuples.

If $\mathcal E_p$ denotes the intended approximation-error functional, the requested attainment property has the schematic form \[ \exists\,\mathbf V^*\in\mathcal S_M(X)\quad \forall\,\mathbf V\in\mathcal S_M(X),\qquad \mathcal E_p(\mathbf V^*)\leq\mathcal E_p(\mathbf V). \] Here $\mathcal E_p$ is an unresolved placeholder, not a functional defined by the supplied record. The record specifies neither the data being approximated nor the formula for their approximation error, the allowed range of $p$ beyond $p\ne2$, or any restrictions on the dimensions or admissibility of the subspaces. In particular, the displayed family includes $(X,\ldots,X)$; whether the intended segmentation problem excludes this tuple is unspecified. If a restricted admissible family is intended, that family must replace $\mathcal S_M(X)$ in the attainment condition.

Thus the supplied question is to characterize minimizer existence for the intended problem when $p\ne2$, but a fully specified mathematical statement requires the missing objective, input data, admissibility hypotheses, and quantifier scope. The record does not determine whether conditions are sought for a particular data set or uniformly over a class of data sets.

Short English statement

The problem asks exactly when an optimal collection of $M$ closed subspaces exists for Banach-space subspace segmentation with exponent $p\ne2$. The supplied record leaves the approximation error, data, exponent range, and subspace restrictions unspecified, so the displayed attainment condition is only a statement schema requiring human mathematical review.

Sources

• Local UnsolvedMath registry: permanent ID $30002093$, title "Minimizers for Banach-Space Subspace Segmentation," source problem code catalogue entry, collection "Oberwolfach Reports Open Problems." No individual proposer is supplied; the recorded proposed year is 2012.
• Supplied report citation: Applied Harmonic Analysis and Sparse Approximation (2013), Oberwolfach Reports, DOI 10.4171/owr/2012/29, pp. 1759--1843. https://doi.org/10.4171/owr/2012/29. The background describes these reports as documenting mathematical workshops and their open-problem discussions. The citation's displayed year is 2013, whereas the DOI and recorded proposed year refer to 2012; their relationship is not explained in the record.
• Supplied literature reference: Aldroubi et al., "review of subspace segmentation." https://onlinelibrary.wiley.com/doi/10.1155/2013/417492. The record supplies no fuller bibliographic details or precise supporting passage.
• Supplied literature assessment, checked on 2026-08-21: the record labels the problem open and states that necessary and sufficient existence conditions are known for $p=2$, while the cited survey and subsequent literature do not provide a general characterization for $p\ne2$. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. The supplied background explicitly calls for independent verification of time-sensitive status and cited claims.