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A note on the affine plank conjecture

Egor Bakaev, Amir Yehudayoff

Abstract

In 1951, Bang posed the affine plank conjecture, which remains open: If a convex body in $\mathbb{R}^d$ is covered by planks, then the total relative width of the planks is at least one. We prove a lower bound of $2/(1+\sqrt{d})$ for this total relative width. The best previously known lower bound was $2/(1+d)$.

A note on the affine plank conjecture

Abstract

In 1951, Bang posed the affine plank conjecture, which remains open: If a convex body in is covered by planks, then the total relative width of the planks is at least one. We prove a lower bound of for this total relative width. The best previously known lower bound was .
Paper Structure (2 sections, 6 theorems, 28 equations, 1 figure)

This paper contains 2 sections, 6 theorems, 28 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The lower bound

Key Result

Theorem 1

If the planks $P_1, \dots, P_m \subset \mathbb{R}^d$ cover a convex body $K\subset\mathbb{R}^d$ then

Figures (1)

  • Figure 1:

Theorems & Definitions (14)

  • Theorem 1: Tarski's plank problem / Bang's theorem
  • Conjecture 2: Bang's affine plank conjecture
  • Lemma 3
  • proof
  • Claim 4
  • proof
  • Lemma 5
  • Lemma 6
  • proof
  • Lemma 7
  • ...and 4 more