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On valuation deviations and their geometry

David Owen Horace Cutler, Mel Deaton

TL;DR

The paper studies two valuation-based deviations Δ_φ and ρ_φ on convex bodies and the intrinsic pseudometrics they induce. It establishes sharp conditions, via the McMullen decomposition, under which the intrinsic pseudometric is dominated by the deviation and when each quantity becomes a genuine metric. In the 1-homogeneous case, the intrinsic metric aligns with ρ_φ, yielding a length-space structure and guaranteed shortest paths; in the n-homogeneous case, the dual behavior holds with Δ_φ acting as a metric and geodesics characterized by bridging bodies, notably for the volume-based case. The results connect valuation theory with intrinsic metric geometry, providing a framework to study a broad class of valuations through geodesic and interpolation properties.

Abstract

We introduce two valuation-based deviations on convex bodies. Using a construction that allows us to associate to these deviations "intrinsic" pseudometrics, we establish various results which capture information about the underlying valuation in terms of the geometry of their induced deviations.

On valuation deviations and their geometry

TL;DR

The paper studies two valuation-based deviations Δ_φ and ρ_φ on convex bodies and the intrinsic pseudometrics they induce. It establishes sharp conditions, via the McMullen decomposition, under which the intrinsic pseudometric is dominated by the deviation and when each quantity becomes a genuine metric. In the 1-homogeneous case, the intrinsic metric aligns with ρ_φ, yielding a length-space structure and guaranteed shortest paths; in the n-homogeneous case, the dual behavior holds with Δ_φ acting as a metric and geodesics characterized by bridging bodies, notably for the volume-based case. The results connect valuation theory with intrinsic metric geometry, providing a framework to study a broad class of valuations through geodesic and interpolation properties.

Abstract

We introduce two valuation-based deviations on convex bodies. Using a construction that allows us to associate to these deviations "intrinsic" pseudometrics, we establish various results which capture information about the underlying valuation in terms of the geometry of their induced deviations.

Paper Structure

This paper contains 7 sections, 19 theorems, 160 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Organization of the paper
  3. Preliminaries and definitions
  4. Valuations
  5. (Semi-)metric geometry
  6. Primary results
  7. Existence of shortest paths

Key Result

Theorem 1.1

Let $\phi\in\mathrm{Val}$ be $k$-strictly monotone and write its McMullen decomposition with each $\phi_i \in \textup{Val}$$i$-homogeneous. Then if and only if $\phi$ has no $1$-homogeneous component, i.e. $\phi_1 \equiv 0.$

Figures (10)

  • Figure 1: A diagonal hyperplane slices a cap of positive area from $L$ while avoiding $K$; the cap lies in $L\setminus K$.
  • Figure 2: With respect to a semimetric, one side of a triangle (i.e. the distance between two vertices) may be strictly greater than the sum of the lengths of the other sides.
  • Figure 3: Two partitions $P, Q$ of $[0,1]$ find a mutual refinement in their union. Additivity of length in a metric space then forces the length computed over $P$ and $Q$ to correspond with it computed over $P \cup Q$.
  • Figure 4: The distance between endpoints compared to a rectifiable path in a semimetric space. The length of the path may be strictly smaller than the endpoint distance, i.e. it may provide a "shortcut".
  • Figure 5: Convex interpolation between nested $K, L \in \mathcal{K}^n_k$ both stays in $\mathcal{K}^n_k$ and is monotone with respect to set inclusion.
  • ...and 5 more figures

Theorems & Definitions (40)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 2.1: McMullen's decomposition, alesker2014 Corollary 1.1.7
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Definition 2.4
  • Example 2.5
  • ...and 30 more