# extreme point

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Naturally, Minkowski's theorem can be easily extended in the framework of global NPC spaces; that is, each point from a closed convex set can be written as a convex combination of extremal points. More precisely, we can say that each point belonging to a convex set belongs to the convex hull of the extremal points.
It diverges at the extremal points of temperature where its sign changes from [C.sub.P] > 0 to [C.sub.P] < 0, or vice versa, which leads to phase transitions, as shown in Figure 8.
Iterating this argument, we have an injection [psi] : V(G) [right arrow] [[??].sub.[greater than or equal to]0] x [[??].sub.[greater than or equal to]0] and it is naturally extended to the map from the set of edges, [??] : E(G) [right arrow] [[??].sub.[greater than or equal to]0] x [[??].sub.[greater than or equal to]0] as [??]([v.sub.a][v.sub.b]) = [??]([v.sub.a]) - [??]([v.sub.b]), the line segment with extremal points [psi]([v.sub.a]) and [psi]([v.sub.b]), for any edge [v.sub.a][v.sub.b] of G.
The relative angle is defined as the angle formed by the lines drawn from the extremal points to the centroid, and the horizontal axis.
Wrap a conceptual string around the set of objects and pull it tight; certain ones of the stakes (in mathematical terms, the convex hull of the set) will touch the string, and each of these are extremal points. How many are there?
It is crucial that the four corners are chosen from extremal points, that is, local minima and maxima of the radius.
For a given n [member of] [??] we use the extremal points of Womersley  on the sphere [S.sup.2].
GOTZ, On the distribution of weighted extremal points on a surface in [R.sup.d], d [greater than or equal to] 3, Potential Anal., 13 (2000), pp.

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