Shortest Piecewise-Smooth Scaling–Rotation Curves

Definitions and mathematical statement

For a positive integer $n$, write \[ \operatorname{Sym}^{+}(n)=\{X\in{\mathbb{R}}^{n\times n}:X^{\mathsf T}=X,\ v^{\mathsf T}Xv>0\text{ for every }v\in{\mathbb{R}}^n\setminus\{0\}\}. \] Let $X,Y\in\operatorname{Sym}^{+}(n)$. The source does not specify a dimension or a range of dimensions for the intended assertion. A continuous piecewise-smooth curve joining these matrices can conventionally be represented by $\gamma:[0,1]\to\operatorname{Sym}^{+}(n)$, with $\gamma(0)=X$, $\gamma(1)=Y$, and a finite partition on whose pieces the curve is smooth; derivatives need not agree at partition points. This convention describes the ambient curves but does not supply the missing scaling--rotation structure.

The supplied problem asks whether the scaling--rotation advantage for smooth curves persists for piecewise-smooth curves, and asks for algebraic conditions under which minimal piecewise-smooth scaling--rotation curves are shortest among all such curves.

The following is only a conditional formulation of the shortestness part. Once a comparison class $\mathcal A(X,Y)$, its length functional $L$, and the intended class $\mathcal M_{\mathrm{SR}}(X,Y)\subseteq\mathcal A(X,Y)$ of minimal piecewise-smooth scaling--rotation curves have been specified, characterize algebraically those $\gamma\in\mathcal M_{\mathrm{SR}}(X,Y)$ satisfying \[ L(\gamma)=\inf_{\eta\in\mathcal A(X,Y)}L(\eta). \] The source does not define these classes, the length functional, the meaning of minimality, or the scaling--rotation advantage. In particular, it does not determine whether the comparison class consists of all piecewise-smooth curves in $\operatorname{Sym}^{+}(n)$ or only curves with a specified scaling--rotation representation. Consequently, the displayed condition records the intended global shortestness question without supplying missing hypotheses or asserting existence of a minimizer. The advantage question cannot be made precise from the supplied record alone.

Short English statement

For symmetric positive-definite matrices $X$ and $Y$, determine whether the stated scaling--rotation advantage extends from smooth to piecewise-smooth curves, and give algebraic criteria for the intended minimal scaling--rotation curves to be globally shortest. The source leaves the advantage, curve classes, length, and meaning of minimality unspecified, so this is a qualified statement-only draft requiring mathematical review.

Sources

• Local UnsolvedMath registry: supplied complete record for permanent ID $30003717$, source problem code catalogue entry, title Shortest Piecewise-Smooth Scaling--Rotation Curves, in the Oberwolfach Reports Open Problems collection. Repository registry locations are data/unsolvedmath/problems.json and catalog/problems.json; neither was independently inspected for this draft.
• Supplied report citation: Statistics for Data with Geometric Structure (2019), Oberwolfach Reports, pp. 125--186, DOI 10.4171/owr/2018/3, https://doi.org/10.4171/owr/2018/3. The record gives proposed year 2018 and describes the report as documenting workshop open-problem discussions. No individual proposer is supplied.
• Supplied literature source: Groisser--Jung--Schwartzman, scaling--rotation curves, https://arxiv.org/abs/1602.01187.
• Supplied literature source: Scaling--rotation uniqueness questions, https://arxiv.org/abs/1702.03237.
• Supplied literature source: Oberwolfach Report 3/2018, https://doi.org/10.4171/owr/2018/3.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that low-dimensional smooth scaling--rotation curves are understood in several cases, while the requested algebraic characterization of shortest curves among all piecewise-smooth curves remains open. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment; its status and cited claims require independent verification.
• The supplied statement status is corrected_verified; its accompanying explanation says that notation, typography, and sentence structure were repaired while preserving mathematical content. This is a supplied editorial label, not mathematical verification of this draft.