Incidence Problems for Families of Convex Bodies

Definitions and mathematical statement

Extend the classical sphere-through-points and hyperplane-through-points incidence problems to convex bodies: determine existence and uniqueness of spheres, balls, or hyperplanes simultaneously tangent to, or slicing prescribed volume fractions from, well-separated families of convex bodies.

In a standard Euclidean setting, let $K_1,\ldots,K_m\subset{\mathbb{R}}^d$ be convex bodies, meaning compact convex sets with nonempty interior. Write $\operatorname{vol}_d$ for Lebesgue volume. A hyperplane with an indicated side has the form \[ H(u,t)=\{x:\langle u,x\rangle=t\},\quad H^+(u,t)=\{x:\langle u,x\rangle\ge t\},\quad |u|=1. \] A prescribed-fraction version of one requested incidence problem would solve \[ \operatorname{vol}_d(K_i\cap H^+(u,t)) =a_i\operatorname{vol}_d(K_i)\qquad(1\le i\le m) \] for specified admissible fractions $a_i$. A ball version would instead use $K_i\cap B(c,r)$ with the same kind of prescribed volume constraints. The associated sphere is $\partial B(c,r)$. These equations distinguish the variants but do not assert their solvability for arbitrary data.

For tangency, a hyperplane ordinarily supports each relevant body and meets it; spherical or ball tangency needs an internal/external or containment convention. The source does not state one. It also does not define well-separated, give $m$ relative to $d$, prescribe general position or strict convexity, specify the allowed fractions or tangency signs, or say whether uniqueness is for oriented hyperplanes, unoriented hyperplanes, balls, spheres, or a selected branch of solutions.

Accordingly, the record is a collection of existence-and-uniqueness research questions rather than one fully quantified theorem. The sphere/ball and hyperplane cases, tangency and volume slicing, and existence and uniqueness remain distinct. No particular definition of well-separated, number of bodies, or universal uniqueness claim is selected to fill the omissions.

Short English statement

The source asks for several convex-body analogues of point-incidence problems, using tangency or prescribed-volume cuts by spheres, balls, and hyperplanes. The separation definition, number of bodies, allowed data, and uniqueness conventions are missing.

Sources

• Local UnsolvedMath registry, permanent ID 30000628, code catalogue entry, "Incidence Problems for Families of Convex Bodies." Proposed year 2006; proposer unspecified.
• Konvexgeometrie (2007), Oberwolfach Reports, pp. 3321--3396, DOI 10.4171/owr/2006/56; https://doi.org/10.4171/owr/2006/56. The displayed report year 2007 differs from the DOI and proposed year 2006. This is the sole supplied literature reference.
• Statement status exact; the supplied verification calls the multipart sentence clear and internally complete and says it was retained byte-for-byte. The missing technical hypotheses and variant choices still require review.
• Literature triage checked 2026-08-21: open. It generically calls the item an unresolved problem/conjecture/direction and reports no authoritative complete resolution in the supplied targeted check, without assessing separate variants. Created 2026-08-25; updated 2026-09-16. Current status and source scope require independent verification.
• Registry curation by Ulam AI / UnsolvedMath Contributors, CC BY 4.0; the underlying report retains its supplied provenance. Unreviewed statement-only draft.