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Mixed volumes of zonoids and the absolute value of the Grassmannian

This paper establishes a combinatorial framework linking the mixed volumes of zonoids to the absolute value of the Grassmannian, utilizing polyhedral computations to derive new Minkowski linear inequalities for specific configurations of zonoids in dimensions two and three.

Original authors: Gennadiy Averkov, Katherina von Dichter, Simon Richard, Ivan Soprunov

Published 2026-07-15

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Gennadiy Averkov, Katherina von Dichter, Simon Richard, Ivan Soprunov

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a magical box of building blocks. In the world of geometry, these blocks are called zonoids. They are special shapes that can be built by stacking up many thin, flat sticks (line segments) in different directions. If you stack them just right, you get a zonotope, a chunky, multi-sided polyhedron. If you keep adding more and more tiny sticks, you can smooth the edges out until you get a perfectly round, smooth shape. These smooth shapes are the zonoids.

Now, imagine you want to measure how much "space" these shapes take up when you mix them together. Mathematicians call this mixed volume. It's a bit like asking: "If I take a slice of shape A, a slice of shape B, and a slice of shape C, how big is the resulting 3D chunk?"

For a long time, mathematicians have been trying to write down a perfect rulebook—a list of every possible inequality—that tells us exactly how these mixed volumes relate to each other. Think of it like a rulebook for a game. You know the basic rules (like "you can't have negative volume"), but you want to know every possible move that is legal and every move that is illegal.

The Big Discovery: A New Rulebook for Specific Cases

In this paper, the authors (Gennadiy, Katherina, Simon, and Ivan) built a brand-new, highly specific rulebook for a particular set of players. They looked at a scenario where you have 6 shapes (let's call them $Z_1$ through $Z_6$ ) and you are working in either 2 dimensions (flat shapes like pancakes) or 3 dimensions (solid shapes like dice).

They discovered that for these specific setups, the rulebook isn't just a vague suggestion; it's a rigid, geometric structure they call a cone. This cone is made of "facets," which are like the flat walls of a crystal. Each wall represents a new, strict inequality that must be true for these shapes.

Here is the cool part: These new rules are Minkowski linear. That's a fancy way of saying that if you take two shapes and glue them together (like stacking two pancakes), the rule applies to the stack exactly the same way it applies to the individual pieces. It's a very clean, predictable kind of math.

The "Rigid" Clues

How did they find these rules? They realized that to understand the whole box of shapes, they only needed to look at the most "locked-in" or rigid arrangements of the building sticks.

Imagine you have 6 sticks on a table. If you can wiggle one stick around without changing the overall "vibe" of the arrangement, it's "free." But if a stick is pinned down by so many other sticks that it can't move at all, it's locked. The authors proved that the most extreme, "locked" arrangements of these sticks are the ones that generate the corners of their new rulebook cone.

For the case of 6 shapes in 2D, they found that the rulebook cone has 975 walls (inequalities). That sounds like a lot, but when you account for the fact that swapping the names of the shapes doesn't change the math, there are really only 8 unique types of new rules.

For 6 shapes in 3D, the cone has 130 walls, which boil down to just 2 unique types of new rules.

The "Absolute Value" Connection

The paper also makes a surprising connection to something called the Grassmannian. In math, this is a fancy way of describing all the possible ways to pick a flat sheet (a plane) out of a higher-dimensional space. Usually, these sheets have a "direction" or "orientation" (like a clock hand pointing clockwise).

The authors looked at the absolute value of these sheets. Imagine taking a photo of the Grassmannian, but instead of seeing the direction of the clock hands, you only see how long they are, ignoring whether they point left or right. They proved that their new rulebook for zonoids is exactly the same as the rulebook for these "direction-less" sheets. It's like finding out that the rules for stacking your building blocks are identical to the rules for measuring the size of shadowy, direction-less planes.

What They Didn't Find (And What They Ruled Out)

It's important to know what this paper doesn't say.

It's not a complete rulebook for everyone: The authors explicitly state that for larger numbers of shapes (like 8 shapes in 2D), this neat, rigid cone structure breaks down. They proved that for 8 shapes, the rulebook is no longer a simple polyhedron with flat walls; it becomes a smooth, curved shape that is much harder to describe. So, their "perfect list" only works for the specific cases of 4, 6, and 6 shapes in 2D and 3D.
Symmetry matters: They found that some of their new rules only work if the shapes are perfectly symmetrical (like a circle or a square). If you use a weird, lopsided shape, some of these rules might fail. They showed a specific example where a rule that works for symmetrical shapes breaks for a general, asymmetrical one.

How Sure Are They?

The authors are very confident about the specific cases they solved. They didn't just guess; they used a powerful computer program called SageMath to do the heavy lifting.

For the cases of 4 shapes in 2D, 6 shapes in 2D, and 6 shapes in 3D, they proved that the rulebook is a polyhedral cone (a shape with flat walls) and they listed the exact number of walls and generators.
For the case of 8 shapes in 2D, they proved that the rulebook is not a polyhedron. They used computer simulations to show that the boundary of the shape curves, meaning you can't describe it with a simple list of flat inequalities anymore.

The Takeaway

This paper is like finding a treasure map for a specific island. The authors mapped out the exact coastline (the inequalities) for islands with 6 beaches (shapes) in 2D and 3D. They found that the coastline is made of straight, sharp cliffs (polyhedral). But they also warned that if you try to map an island with 8 beaches, the coastline becomes a smooth, curving beach that doesn't fit the same simple map.

They used a mix of old-school geometry, clever combinatorial tricks (counting the "locked" stick arrangements), and modern computer power to crack the code for these specific sizes. While they haven't solved the problem for every possible number of shapes, they've given us a crystal-clear view of what happens when we have exactly 6 shapes, and they've shown us exactly where the map stops working.

Technical Summary: Mixed Volumes of Zonoids and the Absolute Value of the Grassmannian

Problem Statement
The paper investigates the system of inequalities governing the mixed volumes of $n$ zonoids in $\mathbb{R}^d$ . While classical geometric inequalities (such as the Aleksandrov-Fenchel inequality) provide fundamental constraints, they do not constitute a "complete system" for $n > 2$ bodies in most dimensions. A complete system is defined here as a set of inequalities that exhaustively describes the relations between the functionals involved. The authors focus specifically on Minkowski multilinear inequalities, which are homogeneous polynomial inequalities of degree $n/d$ that are linear in each zonoid with respect to Minkowski addition.

The core problem is to characterize the pure mixed volume configuration space $PV(n, \mathcal{Z}_d)$ , which consists of vectors of mixed volumes $V(Z_{i_1}, \dots, Z_{i_d})$ for all subsets of size $d$ from a collection of $n$ zonoids. The authors aim to determine the facets of the conic hull $C_{n,d}$ generated by the monomial map applied to these configurations.

Methodology
The authors employ a framework linking convex geometry, combinatorics, and algebraic geometry:

Reduction to Line Segments: Due to the Minkowski multilinearity of the map $\Phi$ (which sends a collection of zonoids to the vector of monomials of their mixed volumes), the verification of inequalities for general zonoids reduces to verifying them for line segments. Every zonoid is a limit of zonotopes (Minkowski sums of segments), and mixed volumes are continuous.
Projective and Combinatorial Viewpoint: An $n$ -tuple of segments is identified with an $n$ -tuple of vectors in $\mathbb{R}^d$ , which corresponds to a configuration of points in the real projective space $\mathbb{RP}^{d-1}$ . The problem is thus translated into studying the image of these point configurations under a "partition monomial map" $\phi$ .
Connection to the Grassmannian: The authors establish that the cone $C_{n,d}$ is the conic hull of the absolute value of the Grassmannian, $\text{Abs}(\text{Gr}_{d,n})$ . This set consists of the absolute values of the Plücker coordinates of $d$ -planes in $\mathbb{R}^n$ . Consequently, finding inequalities for mixed volumes of zonoids is equivalent to finding facet inequalities for the conic hull of $\text{Abs}(\text{Gr}_{d,n})$ .
Rigid Configurations: A key theoretical tool is the concept of rigid configurations of points in projective space. A configuration is "rigid" if every point is "locked" (lying on at least $d$ distinct hyperplanes formed by the other points). The authors prove that for specific cases, the generators of the cone $C_{n,d}$ arise exclusively from these rigid configurations.
Polyhedral Computations: For specific small values of $(n, d)$ , the authors utilize computer-assisted polyhedral computations (using SageMath) to compute the generators and facets of the cone $C_{n,d}$ .

Key Results

Theoretical Framework: The paper proves that for $(n, d) = (4, 2)$ $(n, d) = (4, 2)$ , $(6, 2)$ $(6, 2)$ , and $(6, 3)$ $(6, 3)$ , the cone $C_{n,d}$ $C_{n, d}$ is generated by the images of rigid point configurations.
- For $d=2$ , rigid configurations in $\mathbb{RP}^1$ are characterized by specific multiplicities of points.
- For $d=3$ , rigid configurations in $\mathbb{RP}^2$ involve specific collinearities and double points.
Case $(n, d) = (4, 2)$ : The cone $C_{4,2}$ is a polyhedral cone in $\mathbb{R}^3$ with 3 facets. The authors recover the three Plücker-type inequalities previously introduced in [3], which form a complete system for the "purely mixed" areas of four zonoids in the plane.
Case $(n, d) = (6, 2)$ : The cone $C_{6,2}$ $C_{6, 2}$ is a polyhedral cone in $\mathbb{R}^{15}$ $R^{15}$ with 975 facets. Under the action of the symmetric group $S_6$ $S_{6}$ , these facets fall into 8 orbits. The authors provide explicit coefficients for these 8 types of inequalities.
- Observation: Unlike the $(4,2)$ case, these "best" Minkowski multilinear inequalities do not form a complete system for the general mixed volumes of centrally symmetric bodies; they describe a subset of the full space.
- Symmetry vs. Generality: The authors identify inequalities (specifically Type 3) that hold for centrally symmetric bodies (zonoids in $\mathbb{R}^2$ ) but fail for general convex bodies, highlighting a discrepancy between the zonoid and general convex body cases for $n=6$ .
Case $(n, d) = (6, 3)$ : The cone $C_{6,3}$ is a polyhedral cone in $\mathbb{R}^{10}$ with 130 facets (2 orbits under $S_6$ ). These inequalities are derived from Grassmann-Plücker relations and the triangle inequality.
Non-Polyhedrality: The paper demonstrates that the polyhedral nature of $C_{n,d}$ is not universal. For $d=2$ and even $n \ge 8$ , the cone $C_{n,2}$ is not polyhedral. The authors prove this by showing that the cone contains a continuous family of quadratic curves (arising from 1-parameter families of rigid configurations) that form a non-polygonal boundary in a 2D projection.

Significance and Claims
The paper claims to provide a systematic combinatorial framework linking the study of mixed volumes of zonoids to the absolute value of the Grassmannian. Its primary contributions are:

New Families of Inequalities: The derivation of new, Minkowski linear inequalities for $n=6$ in dimensions $d=2$ and $d=3$ .
Computational Discovery: The paper notes that, to the best of the authors' knowledge, this is the first work in convex geometry where geometric inequalities were determined using computer-assisted tools. This underscores a shift from "hand-derived" proofs to computational methods for complex cases.
Structural Insight: The identification of a transition from polyhedrality to non-polyhedrality as $n$ increases (specifically at $n=8$ for $d=2$ ), suggesting that finding complete systems of inequalities without computer assistance becomes infeasible for most choices of $(n, d)$ .
Distinction of Classes: The work highlights that the system of inequalities for zonoids (centrally symmetric bodies) differs from that of general convex bodies for $n \ge 6$ , a phenomenon not present in the $n=4$ case.

The authors remain modest regarding the general case, noting that describing the generators of $C_{n,d}$ for general $(n, d)$ remains an open problem and that the complexity of the cone structure increases rapidly with $n$ and $d$ .