An exact FullCP prediction region can be star-shaped and still nonconvex, according to an analytic example in an arXiv preprint. In ordinary terms, the region can keep a straight path from one shared center to every point it contains, while still having an inward bend that leaves some points between two included points outside the region.
That distinction changes more than the appearance of a boundary. The analytic example shows that convexification changes both the region’s radial geometry, which tracks its distance from the center by direction, and its Hausdorff geometry, a shape-to-shape distance. A separate constructed counterexample shows that convexity of the candidate score alone does not ensure a connected FullCP region.
The geometry depends on the score
The central geometric result is conditional. For the empirical energy-form pairwise score, comparison differences reduce to sets defined by pairwise dissimilarity thresholds. When the stated symmetry, diagonal, attainment and convexity assumptions hold, these regions combine into an exact star-shaped prediction region around a shared minimizer, the point selected by minimizing the relevant average dissimilarity. The result does not claim that every FullCP region has this form.
Power distance narrows the overlap between the geometric and scoring interpretations. In this specialization, the deterministic geometry is associated with exponents of at least 1, while the conventional energy score is strictly proper for exponents above 0 and below 2. Both statements apply together only from 1 inclusive to below 2. The propriety statement is made for an appropriate finite-moment class.
In the univariate fixed-bin empirical-CRPS case at exponent 1, every nontrivial FullCP region is a nonempty closed interval. It is compact, meaning bounded as well as closed, when there are at least two observations. With one observation, the region becomes the whole real line. That conclusion is specific to this construction.
A boundary that can be checked
The proposed reconstruction uses directions radiating from the shared center. Along each direction, the method records where each comparison set ends, then sorts those exit points. The exact FullCP boundary is the exit at the order-statistic rank set by the required vote threshold. In effect, a multivariate boundary becomes a series of one-dimensional calculations, one for each direction. This representation requires common-center star-shaped comparison sets.
For the power-distance case, the unconditional certification range is narrower: at least two observations and an exponent strictly between 1 and 2. Within that range, the comparison exits and the exact radial function have deterministic Lipschitz bounds. That means the certificate puts a fixed limit on how quickly the boundary can move as direction changes. The tightened global constant is no greater than the old one.
Those bounds support a finite-direction certificate. Give each sampled direction a certified interval for its exit point, and cover the unit sphere with finitely many directions. The resulting lower and upper radial envelopes sandwich the exact region. Their gap, along with the corresponding same-center Hausdorff discrepancies, is bounded by the directional interval width plus twice the tightened Lipschitz constant multiplied by the cover’s radius. This is a guarantee built from directional brackets and the spacing of the cover.
A limited numerical test
The numerical assessment used 160 independent two-dimensional clouds. It covered two observation counts, 10 and 25, four prespecified designs, power exponents 1.1, 1.5 and 1.9, and vote thresholds of 0.10, 0.20 and 0.50. The 40 clouds from the initial smoke stage were retained. Across the expanded settings, the summaries included 1,440 cloud, exponent and vote-threshold rows, but those repeated settings were not independent.
Across 480 cloud-exponent analyses, the tightened-to-old certificate ratio had a median of 0.966, with an interquartile range of about 0.941 to 0.976. That corresponds to a descriptive median tightening of 3.39%, and the tightened bound was no larger in every analysis. These are descriptive comparisons without inferential significance.
At 256 directions, the operational envelope-width target was reached in 128 of 1,440 configurations, or 8.89%, with the old certificate, and in 165 of 1,440, or 11.46%, with the tightened certificate. Neither certificate reached the target at 128 directions or fewer. The tightened envelope was smaller in all 7,200 analysis-by-direction-count comparisons. The target therefore remained uncommon at the tested resolutions.
Nonconvexity was frequently detectable in the same grid. Cellwise robust-witness rates ranged from 40.0% to 81.2%, and 155 of the 160 independent clouds had at least one robust witness somewhere among the prespecified configurations. Conditional on having a witness, however, the normalized mesh-based radial gap had a median of 0.00137, a 90th percentile of 0.00626 and a maximum of 0.0177. The authors characterize the pattern as frequent detection with typically small normalized gaps. A result marked NONE DETECTED is not proof of convexity.
What the evidence does not show
The scope is narrower than the headline finding may suggest. The common-center conclusion depends on the score and explicit assumptions, and the numerical work covers only the stated two-dimensional designs and grid. The paper therefore does not establish that generic FullCP regions are connected or star-shaped. A companion Lean 4 development is reported to machine-check the main early geometric results, while later radial reconstruction, adaptive-bin, exchangeability and coverage questions remain outside its present scope.
Paper data and sources
Original title: Common-Center Geometry and Certified Radial Reconstruction for Energy-Form Full Conformal Regions
Authors: Yiheng Feng
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: Not available
Original paper · Full text