---
title: 'Radon Polytope: Convex & Probabilistic Insights'
url: https://www.emergentmind.com/topics/radon-polytope
type: topic
---

# Radon Polytope: Convex & Probabilistic Insights

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" [2507.05449], 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 [2507.05449].

## 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 [2507.05449].

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**
\[
F(A,B)=\operatorname{conv}\{e_i-e_j: i\in A,\ j\in B\},
\]
where \((A,B)\) is any ordered disjoint pair of subsets of \([N]\), and \(\{e_i\}\) is the standard basis of \(\mathbb R^N\) [2507.05449].

Given a point set \(X=\{x_1,\dots,x_N\}\subset\mathbb R^d\) in affine general position, let
\[
f:\mathbb R^N\to\mathbb R^d,\qquad e_i\mapsto x_i.
\]
Then
\[
V=\ker(f)\cap A
\]
is a linear subspace of \(A\) of codimension \(d\). The **Radon polytope** of \(X\) is defined by
\[
R(X)=P_N\cap V
=\left\{\alpha\in A:\sum_i \alpha_i\le 2,\ \sum_i \alpha_i x_i=0\right\}.
\]
This construction packages all affine dependencies compatible with \(X\) into a single convex body [2507.05449].

The significance of the construction is structural rather than merely representational. The polytope \(P_N\) is universal in \(N\), while the geometry of the particular configuration \(X\) enters only through the linear section by \(V\). This isolates combinatorics in \(P_N\) and geometry in the subspace \(V\).

## 2. Correspondence between faces and Radon partitions

Radon’s theorem is used in the form that any nonzero affine dependence \(\alpha\in A\) satisfying
\[
\sum_i \alpha_i x_i=0
\]
determines a partition by signs:
\[
A=\{i:\alpha_i>0\},\qquad B=\{j:\alpha_j<0\},
\]
and the convex hulls of the two parts meet [2507.05449]. In the complete Radon polytope \(P_N\), such an \(\alpha\) lies on the face \(F(A,B)\).

The paper states the following equivalence for disjoint \(A,B\subseteq [N]\):

- \((A,B)\) is a Radon partition of \(X\);
- \(F(A,B)\cap V\neq \varnothing\);
- the partition cone
  \[
  C(A,B)=\operatorname{cone}\{e_i-e_j:i\in A,\ j\in B\}
  \]
  satisfies
  \[
  C(A,B)\cap V\neq \{0\}.
  \]

As a consequence, the nonempty proper faces of \(P_N\cap V\) are exactly those of the form \(F(A,B)\cap V\), and there is a bijection
\[
\{\text{proper faces of }R(X)\}\longleftrightarrow \{\text{Radon partitions of }X\}.
\]
This is the core encoding theorem of the framework [2507.05449].

A plausible implication is that many combinatorial invariants of Radon partitions can be reinterpreted as face-enumeration problems for sections of \(P_N\). 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 \(X\) to consist of \(N\) independent samples from the \(d\)-dimensional normal distribution \(N(0,I_d)\). In this setting, the paper shows that choosing \(X\) in this way is equivalent, for the induced subspace
\[
V=\ker\!\left(\sum \alpha_i x_i\right),
\]
to choosing a uniformly random \((N-d-1)\)-plane in \(A\) [2507.05449].

For a fixed partition \((A,B)\) with \(|A|=m\) and \(|B|=n\), the probability that it is a Radon partition is
\[
P_d(A,B)
=\mathbb P\{C(A,B)\cap V\neq \{0\}\}
=2\sum_{\substack{i\ge 1\\ i\ \mathrm{odd}}} v_{d+i}(C(A,B)),
\]
where \(v_k(\cdot)\) are the **conic intrinsic volumes** [2507.05449].

By isometry, these intrinsic volumes depend only on the pair \((m,n)\), not on the specific labels of the partition. Writing
\[
v_k(m,n)=v_k(C(A,B)),
\]
the paper obtains the master formula
\[
P_d(A,B)=2\sum_{\substack{i\ge 1\\ i\ \mathrm{odd}}} v_{d+i}(m,n).
\]

In practical calculations, the \(v_k(m,n)\) are computed by inclusion–exclusion on the partition cone and by reduction to the family of regular-simplex cones \(C_n(r)\) studied by Kabluchko–Zaporozhets [2507.05449]. 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 \(X\) to a random linear section \(V\). 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 [2507.05449].

For \(d=1\), an order-statistics argument yields
\[
P_1(A,B)=1-\frac{2}{\binom{m+n}{m}},
\]
and hence
\[
v_0(m,n)=\frac{1}{\binom{m+n}{m}}.
\]
This is one of the cleanest expressions in the framework.

For the case \(m=1\), if \(A=\{n+1\}\) and \(B=[n]\), then
\[
C(A,B)\cong C_n(1),
\]
the cone over an \((n-1)\)-simplex of correlation \(r=1\). The general formula cited in the paper gives
\[
v_k(1,n)
=\binom{n}{k}\; g_k\!\left(-\frac{1}{k+1}\right)\; g_{n-k}\!\left(\frac{1}{k+1}\right),
\]
where \(g_\ell(r)\) are explicit arcsine-integral functions; for example,
\[
g_2(r)=\frac14+\frac{1}{2\pi}\arcsin\!\left(\frac{r}{1+r}\right).
\]

For \(m=2\), the paper states that an inclusion–exclusion argument gives, for \(k=1,\dots,n+1\),
\[
v_k(2,n)
=\binom{n}{k-1}\,g_{k-1}^-\,g_{n-k+1}^+
+2\Bigl[\binom{n}{k}g_{n-k}^+ - \binom{n+1}{k}g_{n-k+1}^+\Bigr]\,g_k^-
\]
plus one more term involving \(g_{k-1}^-\), where
\[
g_\ell^\pm=g_\ell\!\left(\pm \frac{1}{k+1}\right).
\]
The text emphasizes that, although lengthy, these \(g\)-values remain elementary arcsine-integrals [2507.05449].

For the maximal index \(k=m+n-1\), one obtains the compact formula
\[
v_{m+n-1}(m,n)
=\sum_{i=0}^{m-1} (-1)^i \binom{m}{i}\;
g_{n+i}\!\left(-\frac{1}{m+n}\right).
\]
The paper describes this as a surprisingly compact inclusion–exclusion formula [2507.05449].

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 \(X\) is equivalent to asking which faces occur in \(R(X)\)** [2507.05449]. This shifts several open problems into a convex-geometric language.

For **Radon partitions with tolerance**, the problem is to determine the smallest \(t(d)\) such that any \(N\ge t(d)\) points in \(\mathbb R^d\) admit a Radon partition \((A,B)\) that remains Radon after deletion of any single point. In the Radon-polytope formulation, this becomes the existence of a face \(F(A,B)\) of dimension \(N-d-2\) that contains **every** one of its \(N\) sub-ridges. The resulting criterion is therefore a face-enumeration condition on linear sections \(P_N\cap V\) [2507.05449].

For **Reay’s relaxed Tverberg conjecture** in the case \(r=3\), \(k=2\), the question is the minimal
\[
N=T(d,3,2)
\]
such that any \(N\)-point set can be partitioned into three parts \((A,B,C)\) so that each of the three convex-hull pairs intersects. The paper states that this is equivalent to requiring the three faces
\[
F(A,B),\qquad F(A,C),\qquad F(B,C)
\]
of \(P_N\) to meet the random subspace \(V\) simultaneously [2507.05449].

The framework suggests two directions: a **triple-kinematic** generalization of the conic formula, or a contradiction argument proving nonexistence of a subspace \(V\) of the required dimension for \(N=2d+2\) [2507.05449]. 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
\[
R(X)=P_N\cap \ker\!\left(\sum \alpha_i x_i\right)
\]
is a single convex-geometric object whose face lattice encodes every Radon partition of \(X\) [2507.05449]. In the Gaussian model, it becomes a random polytope in \(A\), uniformly distributed among all \((N-d-1)\)-planes. Under this identification, conic-kinematic formulas transform existential combinatorics into exact integral formulas, and in small cases into closed-form arcsine-integrals [2507.05449].

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.

Source: https://www.emergentmind.com/topics/radon-polytope