Circle Free Convex Bodies Under Minkowski Addition

Definitions and mathematical statement

Under the usual convention, a convex body in ${\mathbb{R}}^3$ is a compact convex subset with nonempty interior; write $\mathcal K({\mathbb{R}}^3)$ for this class. For subsets $A,D\subseteq{\mathbb{R}}^3$, their Minkowski sum is \[ A+D=\{a+d:a\in A,\ d\in D\}. \] For $c\in{\mathbb{R}}^3$ and $r>0$, define the closed Euclidean ball \[ B(c,r)=\{x\in{\mathbb{R}}^3:\|x-c\|_2\le r\}. \] The supplied problem asks whether adding a ball to any circle-free convex body in ${\mathbb{R}}^3$ preserves circle-freeness. However, the supplied record does not define circle-free. Let $\operatorname{CF}(K)$ serve only as a placeholder for that intended, presently unspecified property. Under the closed, positive-radius ball convention above, the proposed assertion has the form \[ \forall K\in\mathcal K({\mathbb{R}}^3)\ \forall c\in{\mathbb{R}}^3\ \forall r>0,\qquad \operatorname{CF}(K)\Longrightarrow\operatorname{CF}\left(K+B(c,r)\right). \] This is a formulation template, not a fully specified mathematical assertion: the intended definition of circle-freeness must be recovered from the cited material. The source record also does not explicitly specify whether balls are closed or open, or whether radius zero is allowed; the displayed convention therefore requires confirmation.

Short English statement

Does Minkowski addition of a ball preserve circle-freeness for every convex body in ${\mathbb{R}}^3$? The supplied record leaves circle-freeness undefined, so a complete precise statement requires source clarification. The record labels the problem open; this status has not been independently verified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001883$, title Circle Free Convex Bodies Under Minkowski Addition, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2011 and names no proposer.
• Supplied source citation: Discrete Geometry (2012), Oberwolfach Reports, DOI 10.4171/owr/2011/44, pp. 2459--2548. https://doi.org/10.4171/owr/2011/44. This is also the supplied literature source labeled "original OWR report." The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source labeled "circle-free convex bodies": https://www.fields.utoronto.ca/programs/scientific/11-12/discretegeom/talks/hiroshi.pdf. No author or further bibliographic details are supplied for this link.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that known results for planar sets and special bodies do not settle the question for arbitrary circle-free convex bodies. This is supplied dated literature triage derived from the OWR curation, not an independently verified assessment; time-sensitive status and cited claims require independent checking.
• The supplied statement status is corrected_verified; its accompanying verification describes only enclosing TeX notation in mathematical delimiters without changing mathematical content. It supplies no definition of circle-freeness.