Covering a plane convex body by four homothetical copies with the smallest positive ratio
✍ Scribed by Marek Lassak
- Publisher
- Springer
- Year
- 1986
- Tongue
- English
- Weight
- 498 KB
- Volume
- 21
- Category
- Article
- ISSN
- 0046-5755
No coin nor oath required. For personal study only.
✦ Synopsis
The famous conjecture of Hadwiger [5] that any convex body (i.e. a compact convex set with non-empty interior) of Euclidean n-space E" can be covered by 2" smaller positive homothetical copies remains unsolved for n > 2. For n = 2 the answer is affirmative as was proved in [8] by Levi. A natural question emerges about the smallest possible ratio of those four homothetical copies. The answer is given in the third part of this paper where it is proved that any convex body of E 2 can be covered by four homothetical copies with ratio ½x/~. An extreme example is the disk. It cannot be covered by four homothetical copies with a smaller positive ratio than ½v/2. An additional discussion of the smallest possible ratio is presented in the fourth part. The last part contains some corollaries concerning the question on covering of sets of a Minkowski plane with sets of smaller diameter. The first two parts are auxiliary.
1. QUASI-DUAL AND DUAL PARALLELOGRAMS
We call parallelograms P and Q quasi-dual if the sides of P are parallel to the diagonals of Q and if the sides of Q are parallel to the diagonals of P.
This definition can be also expressed using vectors. Namely, parallelograms P and Q are quasi-dual if and only if for some of the possible denotations of the vectors determined by the pairs of parallel sides of P and Q by Pl, P2 and ql, q2, respectively, the following pairs of vectors are parallel: