Covering by dismembered convex discs
نویسندگان
چکیده
منابع مشابه
Covering by Dismembered Convex Discs
Let us consider a great number of convex discs not "too different" from one another about which, apart from their total area, no further data are known. What can be said about the area of the largest square (or any other domain of given shape) which can be covered with the aid of these discs if we are allowed to cut each of them into a given number of suitable pieces? Analogously, we can raise ...
متن کاملA Note on Covering by Convex Bodies
A classical theorem of Rogers states that for any convex body K in n-dimensional Euclidean space there exists a covering of the space by translates of K with density not exceeding n log n+n log log n+5n. Rogers’ theorem does not say anything about the structure of such a covering. We show that for sufficiently large values of n the same bound can be attained by a covering which is the union of ...
متن کاملOn Visibility and Covering by Convex Sets
A set X IR d is n-convex if among any n its points there exist two such that the segment connecting them is contained in X. Perles and Shelah have shown that any closed (n + 1)-convex set in the plane is the union of at most n 6 convex sets. We improve their bound to 18n 3 , and show a lower bound of order (n 2). We also show that if X IR 2 is an n-convex set such that its complement has one-po...
متن کاملCovering convex bodies by cylinders and lattice points by flats ∗
In connection with an unsolved problem of Bang (1951) we give a lower bound for the sum of the base volumes of cylinders covering a d-dimensional convex body in terms of the relevant basic measures of the given convex body. As an application we establish lower bounds on the number of k-dimensional flats (i.e. translates of k-dimensional linear subspaces) needed to cover all the integer points o...
متن کاملCovering space with convex bodies
1. A few years ago Rogers [1] showed that, if K is any convex body in n-dimensional Euclidian space, there is a covering of the whole space by translates of K with density less than nlogn+nloglogn+5n, provided n > 3. However the fact that the covering density is reasonably small does not imply that the maximum multiplicity is also small. In the natural covering of space by closed cubes, the den...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Proceedings of the American Mathematical Society
سال: 1950
ISSN: 0002-9939
DOI: 10.1090/s0002-9939-1950-0042148-8