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A hyperplane Ham Sandwich theorem

M. C. Crabb

Abstract

We give a direct proof of a result due to Karasev (2008), Karasev-Matschke (2014) and Schnider-Soberón (2023). Given $m+1$ Borel probability measures on the space of affine hyperplanes in a real vector space $V$ of dimension $m+1$, there exist a line $L$ through the origin in $V$ and a point $v\in L$ such that at least half of the hyperplanes, as counted by any of the measures, meet or are parallel to each of the two closed rays in $L$ meeting at $v$.

A hyperplane Ham Sandwich theorem

Abstract

We give a direct proof of a result due to Karasev (2008), Karasev-Matschke (2014) and Schnider-Soberón (2023). Given Borel probability measures on the space of affine hyperplanes in a real vector space of dimension , there exist a line through the origin in and a point such that at least half of the hyperplanes, as counted by any of the measures, meet or are parallel to each of the two closed rays in meeting at .
Paper Structure (2 sections, 7 theorems, 20 equations)

This paper contains 2 sections, 7 theorems, 20 equations.

Table of Contents

  1. The Ham Sandwich theorem
  2. A parametrized version

Key Result

Theorem 1

Suppose that $\mu_0,\ldots ,\mu_l$, $l\geqslant 0$, are Borel probability measures on the space $H^*$ of affine hyperplanes in the $(m+1)$-dimensional real vector space $V$. If $l \leqslant m$, then there is a point $[e,x]\in H$ in the total space of the Hopf bundle over $P(V)$ with the property $(* for $j=0,\ldots ,l$.

Theorems & Definitions (19)

  • Theorem 1
  • Corollary 2
  • proof
  • Example 3
  • Lemma 4
  • proof : Proof of Theorem \ref{['main']}
  • Lemma 5
  • proof
  • Remark 6
  • Theorem A.1
  • ...and 9 more