Additive kinematic formulas for convex functions

This paper establishes a functional version of the additive kinematic formula by combining the Hadwiger theorem on convex functions with a Kubota-type formula for mixed Monge-Ampère measures, thereby providing a new explanation for the equivalence of functional intrinsic volume representations and deriving a novel integral geometric formula for mixed area measures.

The Gist

Imagine you are a master chef in a giant, multi-dimensional kitchen. In this kitchen, you have two main types of ingredients: Solid Shapes (like perfect cubes, spheres, and pyramids) and Flavor Curves (mathematical functions that look like hills, valleys, or bowls).

For a long time, mathematicians had a brilliant rulebook for the Solid Shapes. It was called the "Additive Kinematic Formula." Think of it as a recipe for mixing two shapes together. If you take a cube and a sphere, smash them together, and then spin the sphere around in every possible direction, this rulebook tells you exactly how the total "size" (volume, surface area, etc.) of the resulting mixture behaves on average. It's like a magic calculator that says, "No matter how you spin these two shapes, the average result is always a specific combination of their individual sizes."

The Problem:
This rulebook worked perfectly for solid shapes, but nobody knew how to apply it to the Flavor Curves (convex functions). These curves are more abstract; they don't have hard edges like a cube. They are smooth, continuous hills. Mathematicians had a way to measure these curves, but the formulas were messy, like trying to measure a cloud with a ruler. They needed a new rulebook that worked for these smooth hills, just as the old one worked for the solid blocks.

The Solution (The Paper's Discovery):
The authors of this paper, Daniel, Fabian, and Jacopo, have written a new chapter in the rulebook. They successfully translated the "Solid Shape" mixing rule into a "Flavor Curve" mixing rule.

Here is how they did it, using some creative analogies:

1. The "Shadow" Connection (The Legendre Transform)

Imagine you have a 3D sculpture (a solid shape). If you shine a light on it, it casts a shadow on the wall. In math, there is a special trick called the Legendre Transform (or convex conjugate) that turns a solid shape into a smooth curve and vice versa.

• The Analogy: Think of the solid shape as a physical object and the curve as its "shadow" or "reflection" in a different dimension.
• The Breakthrough: The authors realized that if you know how to mix the shadows (the curves), you can figure out how to mix the objects (the shapes). They used this "shadow" trick to translate the messy curve formulas into something manageable.

2. The "Smoothie" Blender (Infimal Convolution)

When you mix two solid shapes, you use the Minkowski Sum (basically, sliding one shape over the other and collecting all the new points).
When you mix two smooth curves, you can't just slide them. Instead, you use a mathematical operation called Infimal Convolution.

• The Analogy: Imagine you have two smooth hills. To mix them, you don't just stack them. Instead, you imagine rolling a ball between them. The new "mixed" hill is the lowest path the ball can take while touching both original hills. It's like blending two flavors into a new, smoother taste.
• The Result: The paper proves that if you take two curves, mix them using this "rolling ball" method, and then spin one of them around randomly, the average result follows a beautiful, predictable pattern (just like the solid shapes did).

3. The "Universal Translator"

Before this paper, there were two different ways to measure these curves:

1. The "Hessian" Way: Looking at how curved the hill is at every single point (very complex, like counting every grain of sand on a beach).
2. The "Mixed Measure" Way: Looking at the overall "weight" of the curve in different directions (like weighing the whole beach at once).

Mathematicians knew these two methods gave the same answer, but the proof was like a tangled knot. The authors used their new mixing formula to untangle the knot. They showed that the "Hessian" way and the "Mixed Measure" way are just two different languages describing the exact same reality. It's like realizing that "Hello" in English and "Bonjour" in French are just different sounds for the same greeting.

Why Does This Matter?

You might ask, "Who cares about mixing mathematical hills?"

• For Geometry: It unifies two huge branches of math. It shows that the rules governing solid blocks and the rules governing smooth curves are actually part of the same family.
• For Real Life: These mathematical tools are used in:
  • Computer Vision: Helping computers understand 3D shapes from 2D images.
  • Economics: Modeling how prices and resources interact (where "curves" represent value).
  • Physics: Understanding how energy fields behave.

In Summary:
The authors took a complex, abstract problem about "mixing smooth hills" and solved it by realizing these hills are just the "shadows" of solid blocks. They built a new bridge between the world of solid shapes and the world of smooth curves, proving that the same elegant laws of mixing apply to both. They didn't just find a new formula; they found a new way of seeing the mathematical universe, showing that the rules of geometry are universal, whether you are dealing with hard blocks or soft, flowing curves.

Technical Summary

1. Problem Statement

The paper addresses a fundamental gap in integral geometry and convex analysis: the extension of classical kinematic formulas from the space of convex bodies ($K^n$, non-empty compact convex subsets of $\mathbb{R}^n$) to the space of convex functions ($\text{Conv}(\mathbb{R}^n; \mathbb{R})$).

• Classical Context: In classical convex geometry, Hadwiger's Theorem characterizes continuous, translation, and rotation-invariant valuations on $K^n$ as linear combinations of intrinsic volumes $V_j$. A central result derived from this is the Additive Kinematic Formula (Theorem 1.2), which expresses the average intrinsic volume of the Minkowski sum $K + \vartheta L$ (integrated over rotations $\vartheta \in SO(n)$) as a sum of products of intrinsic volumes of $K$ and $L$.
• Functional Context: Recent work (e.g., Hug, Colesanti, Mussnig) established a "Functional Hadwiger Theorem," characterizing valuations on convex functions. These valuations are often represented as singular Hessian integrals involving the eigenvalues of the Hessian matrix $D^2v$. However, these integral representations are difficult to manipulate for kinematic formulas.
• The Challenge: The authors aim to derive a functional additive kinematic formula. The primary difficulty lies in the fact that the natural operation on convex functions corresponding to the Minkowski sum of bodies is the infimal convolution (or epi-sum), denoted by $\square$, rather than pointwise addition. Furthermore, the standard representation of functional intrinsic volumes as Hessian integrals is not well-suited for integration over rotation groups.

2. Methodology

The authors employ a sophisticated combination of tools from convex geometry, integral geometry, and the theory of valuations:

1. Duality and Conjugate Measures:

   • The paper utilizes the Legendre–Fenchel transform (convex conjugation) to map between the space of convex functions $\text{Conv}(\mathbb{R}^n; \mathbb{R})$ and the space of super-coercive convex functions $\text{Conv}_{sc}(\mathbb{R}^n)$.
   • They work extensively with mixed Monge–Ampère measures ($MA$) and their conjugate counterparts ($MA^*$). The key insight is that the infimal convolution in the primal space corresponds to the pointwise sum in the dual space (via conjugates), allowing the use of standard measure-theoretic properties.

2. Kubota-Type Formulas:

   • The authors utilize a Kubota-type formula for conjugate mixed Monge–Ampère measures (Lemma 2.4). This relates integrals over the full space to integrals over lower-dimensional subspaces (Grassmannians), which is crucial for reducing the dimensionality of the problem during the proof.

3. Valuation Theory and Characterization:

   • They rely on the Functional Hadwiger Theorem (Theorem 1.3), which states that any continuous, dually epi-translation, and rotation-invariant valuation on convex functions can be represented as a sum of integrals against mixed Monge–Ampère measures with specific radial densities.
   • The proof strategy involves constructing a functional $Z(u, v)$ defined by integrating a valuation over the rotation group. By showing this functional is itself a valuation, they can apply the Hadwiger characterization to determine its structure.

4. Specific Test Functions:

   • To determine the coefficients in the kinematic formula, the authors evaluate the functional on specific "test" functions: $u_t(x) = t|x| + I_{B^n}(x)$ (where $I_{B^n}$ is the indicator function of the unit ball). These functions act as analogs to scaled balls in the functional setting.

3. Key Contributions and Results

A. The Functional Additive Kinematic Formula (Theorem 1.4)

The central result is a new integral geometric formula for convex functions. For $v, w \in \text{Conv}(\mathbb{R}^n; \mathbb{R})$ and a measurable function $\alpha$:
$$\kappa_n \int_{SO(n)} \int_{\mathbb{R}^n} \alpha(|y|) dMA(v + (w \circ \vartheta^{-1})[j], h_{B^n}[n-j]; y) d\vartheta = \sum_{i=0}^j \binom{j}{i} \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} \alpha(\max\{|x|, |y|\}) dMA(w[j-i], h_{B^n}[n-j+i]; y) dMA(v[i], h_{B^n}[n-i]; x)$$

• Significance: Unlike the classical formula where the result is a sum of products of intrinsic volumes, the right-hand side here involves a double integral with a kernel $\alpha(\max\{|x|, |y|\})$. This reflects the non-linear nature of the infimal convolution operation in the functional setting.
• Corollary 4.2: When one of the functions is the indicator function of a convex body, the formula simplifies to a sum of products of functional intrinsic volumes and classical intrinsic volumes, recovering the classical kinematic formula as a special case.

B. Equivalence of Representations (Section 5)

The paper provides a novel explanation for the equivalence between two representations of functional intrinsic volumes:

1. Singular Hessian Integrals: $\int \zeta(|x|) [D^2v(x)]_j dx$.
2. Mixed Monge–Ampère Integrals: $\int \alpha(|x|) dMA(v[j], h_{B^n}[n-j]; x)$.
   By applying the kinematic formula with a specific choice of functions (specifically using the quadratic function $q(x) = |x|^2/2$), the authors derive the integral transform relating the density functions $\zeta$ and $\alpha$ (the operator $R_{n-j}$). This offers a more geometric and structural proof of the existence of these valuations compared to previous analytic proofs.

C. New Formula for Mixed Area Measures (Theorem 1.5)

The authors derive a new integral geometric formula for mixed area measures of convex bodies, which are fundamental in the Brunn-Minkowski theory.

• Novelty: The integration is performed over the group $SO(n-1) \times O(1)$ (rotations fixing a specific axis and a reflection), rather than the full rotation group $SO(n)$.
• Result: The formula relates the mixed area measure of a Minkowski sum $K + \eta L$ to a double integral over the sphere involving the minimum of the absolute values of the last coordinates ($|z_n|$) of the normal vectors.
  $$\int_{SO(n-1)\times O(1)} \int_{S^{n-1}} |z_n|\beta(|z_n|) dS((K+\eta L)[j], B_H^{n-1}[n-1-j], z) d\eta = \dots$$
  This connects the geometry of convex bodies to the functional kinematic formulas via the projection of bodies onto hyperplanes.

D. Inf-Deconvolution (Corollary 4.4)

The paper introduces an analytic version of the Minkowski difference for convex functions, called inf-deconvolution ($u \diamond v$). Using the kinematic formula, they derive a kinematic formula for the functional intrinsic volumes of the inf-deconvolution, involving alternating sums (similar to the inclusion-exclusion principle).

4. Significance and Impact

1. Unification of Geometric and Functional Theories: The paper successfully bridges the gap between classical convex geometry (bodies) and convex analysis (functions). It demonstrates that the rich structure of integral geometry (kinematic formulas) extends naturally to the functional setting, provided the correct operations (infimal convolution) and measures (Monge–Ampère) are used.
2. New Tools for Valuation Theory: The derived formulas provide powerful new tools for computing and characterizing valuations on convex functions. The "mixed" nature of the right-hand side of the kinematic formula suggests new types of geometric invariants that were previously inaccessible.
3. Resolution of Representation Equivalence: By deriving the relationship between Hessian integrals and Monge–Ampère measures via the kinematic formula, the paper offers a deeper geometric understanding of why these two seemingly different definitions of functional intrinsic volumes are equivalent.
4. Applications to Mixed Area Measures: The new formula for mixed area measures (Theorem 1.5) opens up new avenues for studying the surface geometry of convex bodies using techniques from functional analysis and integral geometry, particularly in contexts involving projections and lower-dimensional symmetries.

In summary, this work establishes a robust functional integral geometry, proving that the Hadwiger framework is not limited to sets but extends to functions, thereby unifying the study of intrinsic volumes, mixed volumes, and their functional analogs under a single kinematic umbrella.