Dimension-Free Separation Exponents for Affine and Positive Semidefinite Sets

Definitions and mathematical statement

Under the usual real symmetric interpretation of the supplied notation, let \[ \operatorname{Sym}_n({\mathbb{R}})=\{X\in{\mathbb{R}}^{n\times n}:X^{\mathsf T}=X\},\qquad S_+^n=\{X\in\operatorname{Sym}_n({\mathbb{R}}):v^{\mathsf T}Xv\geq 0\text{ for every }v\in{\mathbb{R}}^n\}, \] where $n\geq 1$ is an integer. An affine subspace of this ambient space has the form $L=A+V$, with $A\in\operatorname{Sym}_n({\mathbb{R}})$ and $V$ a linear subspace of $\operatorname{Sym}_n({\mathbb{R}})$. These definitions explain a conventional interpretation; the supplied excerpt does not explicitly specify the underlying field or ambient matrix space.

Source question. Is there an algebraic-geometric quantity associated with the relevant affine subspace and positive semidefinite cone that provides a dimension-free exponent in their separation inequality?

Unresolved formulation. The supplied record does not state the separation inequality. Consequently, its distance or residual functions, norm, direction of comparison, admissible exponents, domain of validity, and hypotheses on $L$ cannot be specified from the supplied evidence. In particular, it is not stated whether $L\cap S_+^n$ must be nonempty, whether the inequality is local or global, or which quantities its constants may depend on. The term "dimension-free" indicates a desired independence from the matrix size $n$, but the record does not specify whether the exponent must be a single universal number or may depend on an invariant whose values can themselves vary with $n$. No fully quantified inequality is asserted in this draft.

Short English statement

The question asks for an algebraic-geometric quantity that controls a separation exponent for an affine subspace and the positive semidefinite cone independently of matrix dimension. The actual separation inequality and its hypotheses are missing, so a precise quantified formulation requires further source context.

Sources

• Local UnsolvedMath registry, supplied source record: permanent ID $30006208$, exact source problem code catalogue entry, title "Dimension-Free Separation Exponents for Affine and Positive Semidefinite Sets," in the collection "Oberwolfach Reports Open Problems." The record gives proposed year 2025 and no named proposer in its proposer field.
• Supplied report citation: MFO-RIMS Tandem Workshop: Optimization, Theoretical Computer Science and Algebraic Geometry: Convexity and Beyond (2025), Oberwolfach Reports, pp. 405--450, DOI 10.4171/owr/2025/9. https://doi.org/10.4171/owr/2025/9. The supplied background describes these reports as documenting mathematical workshops and their open-problem discussions.
• The original excerpt includes the heading "Alternating Projections in Low Dimensions," Yoshiyuki Sekiguchi, with joint work credited to Hiroyuki Ochiai and Hayato Waki. This preserves the attached attribution without identifying a proposer beyond what the record establishes. The question refers to a preceding observation that is not supplied.
• Supplied literature source: "OWR report." https://doi.org/10.4171/owr/2025/9.
• Supplied literature source: "Alternating projections for the PSD cone." https://doi.org/10.1016/j.jmaa.2024.129070.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and states that known separation exponents depend on dimension or singularity parameters, while the requested algebraic-geometric dimension-free exponent remains unresolved. This is supplied dated curation, not an independently verified literature assessment; the background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that the official Oberwolfach report was checked, the question isolated, and notation repaired. These are supplied verification claims, not verification performed for this draft.