Exponential Maps for Convex Body Geometric Means

Definitions and mathematical statement

The supplied record uses $K_0^n$, $D$, and a convex-body geometric mean $G$, but does not define these objects. Treating them as source-dependent data, the requested existence assertion is \[ \exists E:K_0^n\longrightarrow K_0^n\quad\text{such that}\quad E(\{0\})=D \] and \[ \forall A,B\in K_0^n,\qquad E\!\left(\frac{A+B}{2}\right)=G\left(E(A),E(B)\right). \] Here $E$ is called an "exponential map" in the source. No additional meaning or requirements for that term are supplied.

Under the usual Minkowski interpretation of the arithmetic mean of sets in ${\mathbb{R}}^n$, \[ A+B=\{a+b:a\in A,\ b\in B\},\qquad \frac{A+B}{2}=\left\{\frac{a+b}{2}:a\in A,\ b\in B\right\}. \] This interpretation must be confirmed against the intended source definitions. For the displayed assertion to be well-defined, $\{0\}$ and $D$ must belong to $K_0^n$, the arithmetic mean must remain in $K_0^n$, and $G$ must be defined on all pairs occurring on the right-hand side. The record does not specify the admissible dimensions, the convention for convex bodies, the meaning of the subscript $0$, or the definitions of $D$ and $G$. In particular, a convention requiring convex bodies to have nonempty interior in positive dimension would exclude $\{0\}$. These unresolved qualifications prevent an unconditional, fully specified mathematical formulation.

Short English statement

The question asks for an exponential map on an intended class of convex bodies that sends $\{0\}$ to $D$ and converts arithmetic means into the specified geometric mean $G$. The class, geometric mean, normalization body, dimension range, and any further requirements on an exponential map need clarification.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json: permanent ID $30003007$, source code catalogue entry, title "Exponential Maps for Convex Body Geometric Means." The supplied record attributes the problem to the Oberwolfach Reports Open Problems collection, records proposed year 2015, and supplies no named proposer.
• Supplied report citation: Convex Geometry and its Applications (2016), Oberwolfach Reports, pp. 3179--3269, DOI 10.4171/owr/2015/56. Source URL and supplied literature-source link, labelled "Convex Geometry and its Applications report": https://doi.org/10.4171/owr/2015/56. The background describes these reports as documenting mathematical workshops and their open-problem discussions. The report has not been independently consulted for this draft.
• Literature check date: 2026-08-21. This is supplied dated curation, not an independently verified assessment. The supplied status is "open"; the supplied assessment states that the proposed exponential map remains an open structural question and that no general construction satisfying the stated arithmetic--geometric identity was found. The background calls this a dated literature triage derived from the supplied OWR curation and requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified, with the qualification that only safe notation, typography, and TeX-delimiter cleanup was applied and that hypotheses, quantifiers, and subquestions were preserved. This supplied label does not establish mathematical verification of this draft.