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

Now, for a given convex body $C$ and $\lambda$ ($\approx 1/2$), what is an upper bound for the number of homothetic copies of $C$ (with coefficient of homothety $\lambda$) required to cover $C$? And what other better bounds exist for some special convex objects?

  • $\begingroup$ For the sphere in 3 dimensions, check out mathoverflow.net/questions/98007/… . $\endgroup$ – The Masked Avenger Feb 11 '14 at 0:59
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    $\begingroup$ The problem of the minimum number of smaller homothetic copies of a convex body needed to cover the body is related to the illumination problem, asking for the minimum number of light sources outside the body needed to illuminate the body's surface. (Here the size of the homothetic copies is not specified, just so that the homothety coefficient is positive and smaller than 1). For a reference on this topic, see the article of K. Bezdek, link.springer.com/chapter/10.1007/978-1-4419-0600-7_3 $\endgroup$ – Wlodek Kuperberg Feb 11 '14 at 4:13

This is not an answer, just a reference.

The 2005 book by Peter Brass, William O. J. Moser, János Pach, Research Problems in Discrete Geometry (Springer link), contains a chapter, "Packing and Covering by Homothetic Copies," pp 121-159, that describes the state of the art. It cites many more results on packing than on covering.

One nice covering result is due to Meir and L. Moser:

Any system of cubes with total volume $1$ can be arranged so as to cover a cube of volume $1/(2^d-1)$.

I don't see your specific question addressed in this chapter, but it has four pages of references.


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