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Same average in every direction

Imre Bárány, Gábor Domokos

Abstract

Given a polytope $P\subset R^3$ and a non-zero vector $z \in R^3$, the plane $\{x\in R^3:zx=t\}$ intersects $P$ in convex polygon $P(z,t)$ for $t \in [t^-,t^+]$ where $t^-=\min \{zx: x \in P\}$ and $t^+=\max \{zx: x\in P\}$, $zx$ is the scalar product of $z,x \in R^3$. Let $A(P,z)$ denote the average number of vertices of $P(z,t)$ on the interval $[t^-,t^+]$. For what polytopes is $A(P,z)$ a constant independent of $z$?

Same average in every direction

Abstract

Given a polytope and a non-zero vector , the plane intersects in convex polygon for where and , is the scalar product of . Let denote the average number of vertices of on the interval . For what polytopes is a constant independent of ?
Paper Structure (13 sections, 6 theorems, 19 equations, 5 figures)

This paper contains 13 sections, 6 theorems, 19 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Higher dimensions
  3. Another expression of $A(P,z)$ and proof of Fact \ref{['fact:zono']}
  4. Proof of Lemma \ref{['l:P-P']}
  5. Non-zonotopes with $A(P,z)$ constant
  6. Proof of Theorem \ref{['th:main']}
  7. A related average
  8. Motivation from geology
  9. Primary fracture and the cube
  10. Secondary fracture interpreted as a recursion
  11. The general case
  12. The 2D case
  13. The 3D case

Key Result

Theorem 1.1

Assume $P\subset \mathbb{R}^3$ is a centrally symmetric polytope with generator set $V$. Then $A(P,z)=\lambda$ for every $z \ne 0$ if and only if $P$ is a zonotope with $2\deg v=\lambda$ for every $v\in V$ in the hypergraph $\mathcal{F}.$

Figures (5)

  • Figure 1: The paths from $v$ to $u$ and from $v'$ to $u'$.
  • Figure 2: A half-zonotope which is not a zonotope.
  • Figure 3: A non-zonotope with $A(P,z)=6$.
  • Figure 4: A non-zonotope with $A(P,z)=4$.
  • Figure 5: Example 6.2 with $\ell=3$, $k=4$ and Example 6.3.

Theorems & Definitions (10)

  • Theorem 1.1
  • Theorem 2.1
  • Lemma 2.1
  • Lemma 5.1
  • Claim 6.1
  • Theorem 7.1
  • Definition 8.1
  • Lemma 8.1
  • proof
  • Definition 8.2