Uneven Splitting of Ham Sandwiches

نویسنده

  • Felix Breuer
چکیده

Let μ1, . . . , μn be continuous probability measures on Rn and α1, . . . , αn ∈ [0, 1]. When does there exist an oriented hyperplane H such that the positive half-space H+ has μi(H) = αi for all i ∈ [n]? It is well known that such a hyperplane does not exist in general. The famous ham sandwich theorem states that if αi = 2 for all i, then such a hyperplane always exists. In this paper we give sufficient criteria for the existence of H for general αi ∈ [0, 1]. Let f1, . . . , fn : Sn−1 → Rn denote auxiliary functions with the property that for all i the unique hyperplane Hi with normal v that contains the point fi(v) has μi(H i ) = αi. Our main result is that if Im f1, . . . , Im fn are bounded and can be separated by hyperplanes, then there exists a hyperplane H with μi(H) = αi for all i. This gives rise to several corollaries, for instance if the supports of μ1, . . . , μn are bounded and can be separated by hyperplanes, then H exists for any choice of α1, . . . , αn ∈ [0, 1]. We also obtain results that can be applied if the supports of μ1, . . . , μn overlap.

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Growth Profile and Toxigenicity of Bacillus cereus in Ready-to-eat Food Products of Animal Origin

The growth profile of Bacillus cereus in ready-to-eat (RTE) food products of animal origin was examined under different temperature and incubation conditions. In sandwiches and Kimbab, B. cereus did not grow or exhibited only minimal growth at 4 and 10oC, but it grew rapidly at ambient temperature. In sandwiches, B. cereus did not grow efficiently at 25oC, however, in ham, the main ingredient o...

متن کامل

A foodborne outbreak of gastroenteritis associated with Norwalk-like viruses: first molecular traceback to deli sandwiches contaminated during preparation.

In March 1998, an outbreak of acute gastroenteritis occurred among students at a Texas university. Overall, 125 ill students sought medical care. Case-control studies revealed that illness was significantly associated with eating foods from the university's main cafeteria deli bar on 9 and 10 March. Stool specimens from 9 (50%) of 18 ill students and samples of deli ham showed evidence of Norwa...

متن کامل

White polarization sandwiches: optical elements with non-orthogonal eigenpolarizations

Optical elements with non-orthogonal eigenpolarizations have complex anisotropic properties, which are not yet well understood. As an example of such elements, we studied theoretically the class of white polarization sandwiches. Light passing successively through two identical white sandwiches preserves its polarization state, so that, following the terminology of Berry and Dennis, white polari...

متن کامل

Creating New Sandwiches From Old

Earl is a case-based reasoning system which works in the domain of sandwich recipes. It has a case base of 9,507 cases, each being a sandwich recipe which we scraped from the web. We show how Earl can compose new sandwiches from the ingredients of sandwiches that it retrieves from its case base. On each iteration, it inserts into the sandwich the candidate ingredient which results in greatest c...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:
  • Discrete & Computational Geometry

دوره 43  شماره 

صفحات  -

تاریخ انتشار 2010