Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Given a convex body $C\subset R^d$ and a positive real $\lambda$, any set of the form $\lambda C+x$ = {$~\lambda c+x|c\in C$} for some $x\in R^d$ is called homothetic copy of $C$. The number $\lambda>0$ is called as coefficient of homothety.

Let $C$ be a $d$-cube ($d$-dimensional cube) of side length 1. Now draw another cube $C'$ concentric and homothetic w.r.t. $C$ and having $\lambda= 1/2-\epsilon$ (or equivalently, $C'$ is of side $1/2-\epsilon$).

Now,

1) How many translated copies of $C'$ would be require to cover the annulus obtained between $C'$ and $C$?

2) How many translated copies of $C'$ would be require to cover $C$?

share|improve this question
add comment

1 Answer 1

up vote 4 down vote accepted

$3^d-1$ and $3^d$, resp. Let $C=[0,1]^d$, Consider the $3^d$ points in $C$ all whose coordinates are from the set $\{0,\frac12,1\}$. No translated copy of $C'$ can cover two of these points, hence at least $3^d$ copies of $C'$ are required. On the other hand, a covering by $3^d$ copies of $C'$ (with the requirement that one of them is concentric to $C$) is easy to construct by induction in $d$.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.