The Infona portal uses cookies, i.e. strings of text saved by a browser on the user's device. The portal can access those files and use them to remember the user's data, such as their chosen settings (screen view, interface language, etc.), or their login data. By using the Infona portal the user accepts automatic saving and using this information for portal operation purposes. More information on the subject can be found in the Privacy Policy and Terms of Service. By closing this window the user confirms that they have read the information on cookie usage, and they accept the privacy policy and the way cookies are used by the portal. You can change the cookie settings in your browser.
We generalize the ham sandwich theorem to d+1 measures on Rd as follows. Let μ1,μ2,…,μd+1 be absolutely continuous finite Borel measures on Rd. Let ωi=μi(Rd) for i∈[d+1], ω=min{ωi;i∈[d+1]} and assume that ∑j=1d+1ωj=1. Assume that ωi≤1/d for every i∈[d+1]. Then there exists a hyperplane h such that each open halfspace H defined by h satisfies μi(H)≤(∑j=1d+1μj(H))/d for every i∈[d+1] and ∑j=1d+1μj(H)≥min{1/2,1−dω}≥1/(d+1)...
Suppose that nk points in general position in the plane are colored red and blue, with at least n points of each color. We show that then there exist n pairwise disjoint convex sets, each of them containing k of the points, and each of them containing points of both colors.We also show that if P is a set of n(d+1) points in general position in Rd colored by d colors with at least n points of each...
Set the date range to filter the displayed results. You can set a starting date, ending date or both. You can enter the dates manually or choose them from the calendar.