Generalized Covariograms of Planar Convex Bodies

Definitions and mathematical statement

Let $K\subset{\mathbb{R}}^2$ be a convex body, meaning compact, convex and with nonempty interior. Fix a known real-valued valuation $\varphi$ on the appropriate class of compact convex sets, including the intersection/empty-set conventions needed below. A valuation satisfies \[ \varphi(A)+\varphi(B)=\varphi(A\cup B)+\varphi(A\cap B) \] when the sets are in its domain and $A\cup B$ is convex. Translation invariance means $\varphi(A+t)=\varphi(A)$, and strict monotonicity means $\varphi(A)<\varphi(B)$ for proper inclusion $A\subsetneq B$ in the intended class. Perimeter is the example supplied by the source.

The generalized covariogram is the full function \[ g_{\varphi,K}(x)=\varphi\left(K\cap(K+x)\right),\qquad x\in{\mathbb{R}}^2. \] The question is whether, for every admissible fixed $\varphi$ and planar bodies $K,L$, \[ g_{\varphi,K}=g_{\varphi,L} \quad\Longrightarrow\quad L=K+t\ \text{ or }\ L=-K+t\quad\text{for some }t\in{\mathbb{R}}^2, \] under the usual covariogram meaning of reflections as point reflections. The source does not distinguish this from arbitrary line reflections; that convention must be confirmed. The valuation is the same for both bodies, not another unknown to be reconstructed.

Intersections may be empty, segments or points, so a valuation specified only for full-dimensional bodies needs an extension. For perimeter the usual continuous extension gives twice the length on a segment and zero on a point/empty set, but the record does not state a general boundary-value convention. Continuity, homogeneity, rotation invariance and evenness $\varphi(-A)=\varphi(A)$ are not among the supplied hypotheses and are not added. In particular, point-reflecting the body preserves the data automatically for an even valuation; for a general translation-invariant valuation that symmetry should not be presumed solely from the wording of the recovery class.

The literature triage checked 21 August 2026 reports no authoritative complete resolution of the extracted assertion. It gives no precise valuation subclass or theorem beyond the report. That dated assessment is preserved without an independent current-status conclusion.

Short English statement

Does the full intersection-valuation function determine a planar convex body up to translation and the intended reflection? The valuation is fixed, strictly monotone and translation invariant, with perimeter as the supplied example. Lower-dimensional intersections and the reflection convention require clarification; extra symmetry assumptions are not silently added.

Sources

• Local UnsolvedMath registry, permanent ID 30000630, exact source code catalogue entry; Generalized Covariograms of Planar Convex Bodies. Proposed year 2006, status open; corrected_verified extraction retains the valuation hypotheses, perimeter example and stated recovery equivalence.
• Supplied report citation: Konvexgeometrie (2007), Oberwolfach Reports, DOI 10.4171/owr/2006/56, pp. 3321--3396, https://doi.org/10.4171/owr/2006/56. Title year differs from DOI/proposed year.
• Supplied literature triage checked 2026-08-21 cites the same report, https://doi.org/10.4171/owr/2006/56, and reports no authoritative complete resolution found. The dated OWR-derived curation requests independent verification of time-sensitive status and cited claims.