Radon Polytope: Convex & Probabilistic Insights

• Radon polytope is a convex body defined from affine dependencies that encodes every Radon partition of a finite point set.
• It establishes a deterministic correspondence between its faces and Radon partitions through linear sections of a universal polytope.
• In the Gaussian model, the framework transforms combinatorial geometry problems into probability calculations using conic kinematic formulas and intrinsic volumes.

Searching arXiv for the cited paper and closely related background on conic intrinsic volumes and simplex cones. A Radon polytope is a convex-geometric object associated with a finite point set $X=\{x_1,\dots,x_N\}\subset \mathbb R^d$ in affine general position, constructed so that its face lattice encodes exactly the Radon partitions of $X$. In the framework developed in "Radon Partitions of Random Gaussian Polytopes" (White, 7 Jul 2025), the Radon polytope arises as a linear section of a universal polytope $P_N$ defined in the space of affine dependencies on $N$ labeled points. This perspective yields a deterministic correspondence between faces and Radon partitions, and in the Gaussian model it converts questions about the existence of Radon partitions into probability calculations governed by conic kinematic formulas and conic intrinsic volumes (White, 7 Jul 2025).

1. Deterministic construction

Let $N\ge 2$ and define

$$A=\{\alpha=(\alpha_1,\dots,\alpha_N)\in\mathbb R^N:\sum_{i=1}^N \alpha_i=0\}.$$

This is the $(N-1)$-dimensional space of affine dependencies on $N$ points (White, 7 Jul 2025).

Within $A$, the paper defines the complete Radon polytope

$P_N=\{\alpha\in A:\sum_{i=1}^N \alpha_i\le 2\}.$

Its proper faces are precisely the partition faces

$X$0

where $X$1 is any ordered disjoint pair of subsets of $X$2, and $X$3 is the standard basis of $X$4 (White, 7 Jul 2025).

Given a point set $X$5 in affine general position, let

$X$6

Then

$X$7

is a linear subspace of $X$8 of codimension $X$9. The Radon polytope of $P_N$0 is defined by

$P_N$1

This construction packages all affine dependencies compatible with $P_N$2 into a single convex body (White, 7 Jul 2025).

The significance of the construction is structural rather than merely representational. The polytope $P_N$3 is universal in $P_N$4, while the geometry of the particular configuration $P_N$5 enters only through the linear section by $P_N$6. This isolates combinatorics in $P_N$7 and geometry in the subspace $P_N$8.

2. Correspondence between faces and Radon partitions

Radon’s theorem is used in the form that any nonzero affine dependence $P_N$9 satisfying

$N$0

determines a partition by signs: $N$1 and the convex hulls of the two parts meet (White, 7 Jul 2025). In the complete Radon polytope $N$2, such an $N$3 lies on the face $N$4.

The paper states the following equivalence for disjoint $N$5:

• $N$6 is a Radon partition of $N$7;
• $N$8;
• the partition cone

$N$9

satisfies

$N\ge 2$0

As a consequence, the nonempty proper faces of $N\ge 2$1 are exactly those of the form $N\ge 2$2, and there is a bijection

$N\ge 2$3

This is the core encoding theorem of the framework (White, 7 Jul 2025).

A plausible implication is that many combinatorial invariants of Radon partitions can be reinterpreted as face-enumeration problems for sections of $N\ge 2$4. The paper makes this viewpoint explicit in later applications to tolerance and relaxed Tverberg-type questions.

3. Gaussian randomization and conic kinematic formulas

The probabilistic model takes $N\ge 2$5 to consist of $N\ge 2$6 independent samples from the $N\ge 2$7-dimensional normal distribution $N\ge 2$8. In this setting, the paper shows that choosing $N\ge 2$9 in this way is equivalent, for the induced subspace

$$A=\{\alpha=(\alpha_1,\dots,\alpha_N)\in\mathbb R^N:\sum_{i=1}^N \alpha_i=0\}.$$0

to choosing a uniformly random $$A=\{\alpha=(\alpha_1,\dots,\alpha_N)\in\mathbb R^N:\sum_{i=1}^N \alpha_i=0\}.$$1-plane in $$A=\{\alpha=(\alpha_1,\dots,\alpha_N)\in\mathbb R^N:\sum_{i=1}^N \alpha_i=0\}.$$2 (White, 7 Jul 2025).

For a fixed partition $$A=\{\alpha=(\alpha_1,\dots,\alpha_N)\in\mathbb R^N:\sum_{i=1}^N \alpha_i=0\}.$$3 with $$A=\{\alpha=(\alpha_1,\dots,\alpha_N)\in\mathbb R^N:\sum_{i=1}^N \alpha_i=0\}.$$4 and $$A=\{\alpha=(\alpha_1,\dots,\alpha_N)\in\mathbb R^N:\sum_{i=1}^N \alpha_i=0\}.$$5, the probability that it is a Radon partition is

$$A=\{\alpha=(\alpha_1,\dots,\alpha_N)\in\mathbb R^N:\sum_{i=1}^N \alpha_i=0\}.$$6

where $$A=\{\alpha=(\alpha_1,\dots,\alpha_N)\in\mathbb R^N:\sum_{i=1}^N \alpha_i=0\}.$$7 are the conic intrinsic volumes (White, 7 Jul 2025).

By isometry, these intrinsic volumes depend only on the pair $$A=\{\alpha=(\alpha_1,\dots,\alpha_N)\in\mathbb R^N:\sum_{i=1}^N \alpha_i=0\}.$$8, not on the specific labels of the partition. Writing

$$A=\{\alpha=(\alpha_1,\dots,\alpha_N)\in\mathbb R^N:\sum_{i=1}^N \alpha_i=0\}.$$9

the paper obtains the master formula

$(N-1)$0

In practical calculations, the $(N-1)$1 are computed by inclusion–exclusion on the partition cone and by reduction to the family of regular-simplex cones $(N-1)$2 studied by Kabluchko–Zaporozhets (White, 7 Jul 2025). This places the probability problem within the standard apparatus of conic integral geometry.

The central conceptual step is that the random geometry is transferred from the point configuration $(N-1)$3 to a random linear section $(N-1)$4. This recasts an existential question about convex hull intersections as an intersection probability between a fixed cone and a random subspace.

4. Closed forms and explicit small-parameter cases

The paper derives exact formulas in several regimes and closed forms in some of them (White, 7 Jul 2025).

For $(N-1)$5, an order-statistics argument yields

$(N-1)$6

and hence

$(N-1)$7

This is one of the cleanest expressions in the framework.

For the case $(N-1)$8, if $(N-1)$9 and $N$0, then

$N$1

the cone over an $N$2-simplex of correlation $N$3. The general formula cited in the paper gives

$N$4

where $N$5 are explicit arcsine-integral functions; for example,

$N$6

For $N$7, the paper states that an inclusion–exclusion argument gives, for $N$8,

$N$9

plus one more term involving $A$0, where

$A$1

The text emphasizes that, although lengthy, these $A$2-values remain elementary arcsine-integrals (White, 7 Jul 2025).

For the maximal index $A$3, one obtains the compact formula

$A$4

The paper describes this as a surprisingly compact inclusion–exclusion formula (White, 7 Jul 2025).

These cases show that the general conic-kinematic expression is exact but not uniformly simple: some parameter ranges collapse to closed forms, whereas others still require repeated integration.

5. Reformulation of tolerance and relaxed Tverberg-type questions

A central claim of the paper is that every combinatorial question about which Radon partitions occur in $A$5 is equivalent to asking which faces occur in $A$6 (White, 7 Jul 2025). This shifts several open problems into a convex-geometric language.

For Radon partitions with tolerance, the problem is to determine the smallest $A$7 such that any $A$8 points in $A$9 admit a Radon partition $P_N=\{\alpha\in A:\sum_{i=1}^N \alpha_i\le 2\}.$0 that remains Radon after deletion of any single point. In the Radon-polytope formulation, this becomes the existence of a face $P_N=\{\alpha\in A:\sum_{i=1}^N \alpha_i\le 2\}.$1 of dimension $P_N=\{\alpha\in A:\sum_{i=1}^N \alpha_i\le 2\}.$2 that contains every one of its $P_N=\{\alpha\in A:\sum_{i=1}^N \alpha_i\le 2\}.$3 sub-ridges. The resulting criterion is therefore a face-enumeration condition on linear sections $P_N=\{\alpha\in A:\sum_{i=1}^N \alpha_i\le 2\}.$4 (White, 7 Jul 2025).

For Reay’s relaxed Tverberg conjecture in the case $P_N=\{\alpha\in A:\sum_{i=1}^N \alpha_i\le 2\}.$5, $P_N=\{\alpha\in A:\sum_{i=1}^N \alpha_i\le 2\}.$6, the question is the minimal

$P_N=\{\alpha\in A:\sum_{i=1}^N \alpha_i\le 2\}.$7

such that any $P_N=\{\alpha\in A:\sum_{i=1}^N \alpha_i\le 2\}.$8-point set can be partitioned into three parts $P_N=\{\alpha\in A:\sum_{i=1}^N \alpha_i\le 2\}.$9 so that each of the three convex-hull pairs intersects. The paper states that this is equivalent to requiring the three faces

$X$00

of $X$01 to meet the random subspace $X$02 simultaneously (White, 7 Jul 2025).

The framework suggests two directions: a triple-kinematic generalization of the conic formula, or a contradiction argument proving nonexistence of a subspace $X$03 of the required dimension for $X$04 (White, 7 Jul 2025). This suggests that the Radon-polytope viewpoint is not limited to pairwise convex-hull intersection, but may organize higher-order partition problems as simultaneous incidence constraints for multiple faces.

6. Convex-geometric significance

The paper’s concluding formulation is that

$X$05

is a single convex-geometric object whose face lattice encodes every Radon partition of $X$06 (White, 7 Jul 2025). In the Gaussian model, it becomes a random polytope in $X$07, uniformly distributed among all $X$08-planes. Under this identification, conic-kinematic formulas transform existential combinatorics into exact integral formulas, and in small cases into closed-form arcsine-integrals (White, 7 Jul 2025).

This viewpoint has two notable consequences. First, it gives a unified deterministic and probabilistic language for Radon partitions: deterministic incidence is encoded by faces of a section, while random incidence is encoded by intersection probabilities with a random subspace. Second, it places Radon-type questions in the orbit of conic integral geometry, where intrinsic volumes and kinematic identities provide exact analytic control.

A plausible implication is that the Radon polytope functions as an intermediary between convex-geometric combinatorics and probabilistic asymptotics. The framework does not merely count or certify Radon partitions; it organizes them into a face structure whose geometry is amenable to both exact calculation and reformulation of open problems.