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Normal-sized hypercuboids in a given hypercube

Takashi Hirotsu

TL;DR

The limit of the value of the ratio of the arithmetic mean of the volumes of those hypercuboids (or hypercubes) whose edges are lying on the grid lines or the boundary is found.

Abstract

In a given hypercube, draw grid lines parallel to the edges, and consider all hypercuboids (or hypercubes) whose edges are lying on the grid lines or the boundary. We find the limit of the value of the ratio of the arithmetic mean of the volumes of those hypercuboids (or hypercubes) to the entire volume as the grid spacing becomes smaller.

Normal-sized hypercuboids in a given hypercube

TL;DR

The limit of the value of the ratio of the arithmetic mean of the volumes of those hypercuboids (or hypercubes) whose edges are lying on the grid lines or the boundary is found.

Abstract

In a given hypercube, draw grid lines parallel to the edges, and consider all hypercuboids (or hypercubes) whose edges are lying on the grid lines or the boundary. We find the limit of the value of the ratio of the arithmetic mean of the volumes of those hypercuboids (or hypercubes) to the entire volume as the grid spacing becomes smaller.
Paper Structure (5 theorems, 17 equations)

This paper contains 5 theorems, 17 equations.

Key Result

Theorem 1

In an $n$-dimensional hypercube $H \subset \mathbb R^n,$ the value $q_n$ of the ratio of the volume of a normal-sized hypercuboid to the entire volume is

Theorems & Definitions (11)

  • Definition 1
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Proposition 1
  • proof
  • Theorem 3
  • proof
  • Proposition 2
  • ...and 1 more