Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Valuations on Convex Bodies and Functions

Monika Ludwig, Fabian Mussnig

Abstract

An introduction to geometric valuation theory is given. The focus is on classification results for $\operatorname{SL}(n)$ invariant and rigid motion invariant valuations on convex bodies and on convex functions.

Valuations on Convex Bodies and Functions

Abstract

An introduction to geometric valuation theory is given. The focus is on classification results for invariant and rigid motion invariant valuations on convex bodies and on convex functions.
Paper Structure (34 sections, 47 theorems, 269 equations, 4 figures)

This paper contains 34 sections, 47 theorems, 269 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Basic Properties
  3. $\operatorname{SL}(n)$ Invariant Valuations
  4. The One-dimensional Case
  5. $\operatorname{SL}(n)$ Invariant Valuations on Convex Polytopes
  6. Affine Surface Area
  7. Translation Invariant Valuations
  8. The Canonical Simplex Decomposition
  9. Valuations Vanishing on Orthogonal Cylinders
  10. The Homogeneous Decomposition Theorem
  11. Rigid Motion Invariant Valuations
  12. A Characterization of the Mean Width
  13. Proof of Proposition \ref{['prop_hugo_simple']}
  14. Valuations Invariant under Subgroups of $\operatorname{O}(n)$
  15. An Application of the Hadwiger Theorem
  16. ...and 19 more sections

Key Result

Lemma 2.1

Let $\mathop{\mathrm{\operatorname{Z}}}\nolimits:{\mathcal{K}}^n\to {\mathbb R}$ be a valuation. If $C \in {\mathcal{K}}^n$ is a fixed convex body and for $K \in {\mathcal{K}}^n$, then $\mathop{\mathrm{\operatorname{Z}}}\nolimits_{C}$ is a valuation on ${\mathcal{K}}^n$.

Figures (4)

  • Figure 1: Decomposition of $[v_0,\ldots,v_n]$ into $T_1$ and $T_2$.
  • Figure 2: Support triangle of a convex body $K\in{\mathcal{K}}^2$ with endpoints $x,y\in\partial K$.
  • Figure 3: Canonical simplex decomposition
  • Figure 4: Illustration of $u$ and $\bar{u}$ for the case $k=3$.

Theorems & Definitions (74)

  • Lemma 2.1
  • proof
  • Theorem 3.1: Blaschke
  • Proposition 3.2
  • proof
  • Corollary 3.3
  • Theorem 3.4
  • proof
  • Corollary 3.5
  • Theorem 3.6
  • ...and 64 more