If $C$ meets the interior of $K$ and the dual of $K$ has non-empty interior, then the point can be taken to have a non-zero normal in the interior of the dual of $K$. intersects the affine space Ax y into a hyperplane Hi. The intersection of the Nonnegative Orthant with Complementary Orthogonal Subspaces. If crossed a hyperplane twice, in points xand y, you could shorten by replacing the part of between xand yby the projection of onto the hyperplane. If it intersects an orthant, the intersection is either a point or a straight line segment.
#Check if hyperplan intersects orthant download#
Then $C\cap K$ contains a point with a non-zero normal in the dual of $K$. portion that borders the non-negative orthant, that is, x might have negative coordinates. Diagrammatic Analysis of Interval Linear Equations -Part II: The Two-Dimensional Case and Generalization to n Dimensions Zenon Kulpa download BookSC. crosses each hyperplane x i 0 exactly once. Proposition: Let $C\subseteq \mathbb R^n$ be a compact convex set that meets some closed convex cone $K$. interesting case is therefore when ker B intersects the positive orthant the. have any intersection with the strict negative orthant in Rk. Many results are known on these systems (see 15, 7, 8, 10, 16).
While fedja gave an answer in the comments, here is a different approach that yields a more general answer (the non-negative orthant is self-dual). So p needs to define a hyperplane that puts Zh(xh,eh) on its strictly positive.
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