Consider the corner hypercubes of length o 1 inside a unit hypercube. The 2-dime
ID: 1720109 • Letter: C
Question
Consider the corner hypercubes of length o 1 inside a unit hypercube. The 2-dimensional case is shown in Figure 6.11. Answer the following questions:
(a) Let o = 0.1. What is the fraction of the total volume occupied by the corner cubes
in two dimensions?
(b) Derive an expression for the volume occupied by all of the corner hypercubes of
length o < 1 as a function of the dimension d. What happens to the fraction of the
volume in the corners as d ?
(c) What is the fraction of volume occupied by the thin hypercube shell of width o < 1
as a fraction of the total volume of the outer (unit) hypercube, as d ? For example, in two dimensions the thin shell is the space between the outer square (solid) and inner square (dashed).
Explanation / Answer
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