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https://www.cis.upenn.edu/~cis610/convex1-09.pdf
BASIC PROPERTIES OF CONVEX SETS The answer is yes in both cases. In case 1, assuming thattheaffinespaceE hasdimensionm, Carath´eodory’s Theorem asserts that it is enough to consider convex combinations of m+1 points. In case 2, the theorem of Krein and Milman asserts that a convex set which is also compact is the convex hull of
https://kaniskadam.weebly.com/uploads/4/2/7/7/42773651/chapter4-2017.pdf
CHAPTER 4: Convex Sets and Separation Theorems 1 Convex sets Convexity is often assumed in economic theory since it plays important roles in optimization. The convexity of preferences can be interpreted as capturing consumer’s liking for variety, and the convexity of production set is related to the existence of nonincreasing returns to scale.
http://www.ifp.illinois.edu/~angelia/L7_separationthms.pdf
Lecture 7. Supporting Hyperplane Theorem. Th. Let C ⊆ Rn be a nonempty convex set. Let x0 be such that either x0 ∈ bdC or x0 ∈/ C. Then, there exists a hyperplane passing through x0 and containing the set. C in one of its halfspaces, i.e., there is a vector a ∈ Rn, a 6= 0 , such that. sup.
https://ljk.imag.fr/membres/Anatoli.Iouditski/cours/convex/chapitre_1.pdf
Convex Sets (Basic de nitions and properties; Separation theorems; Characterizations) 1.1 De nition, examples, inner description, algebraic properties 1.1.1 A convex set In the school geometry a gure is called convex if it contains, along with any pair of its points x;y, also the entire segment [x;y] linking the points. This is exactly the de nition
https://www.worldscientific.com/worldscibooks/10.1142/9508
Apr 01, 2015 · Topics under consideration include general properties of convex sets and convex hulls, cones and conic hulls, polyhedral sets, the extreme structure, support and separation properties of convex sets. Lectures on Convex Sets is self-contained and unified in presentation. The book grew up out of various courses on geometry and convexity, taught by the author for more than a decade.
https://link.springer.com/article/10.1007/s10492-010-0021-9
Nov 10, 2010 · V. Klee: Separation and support properties of convex sets—a survey. Control Theory and the Calculus of Variations (A.V. Balakrishnan, ed.). Control Theory and the Calculus of Variations (A.V. Balakrishnan, ed.).Cited by: 2
https://www.researchgate.net/publication/2407147_Separation_Of_Two_Convex_Sets_In_Convexity_Structures
A convexity structure satisfies the separation property S4 if any two disjoint convex sets extend to complementary half-spaces. This property is investigated for alignment spaces, n-ary...Author: Victor Chepoi
https://www.sciencedirect.com/science/article/pii/0022247X73900760
The convex pair (C, D) separates the sets A and B if either A C C and B C D or A C D and B C C. (3) THEOREM. Any two disjoint non-empty convex sets can be separated by a convex pair. Proof.Cited by: 17
https://en.wikipedia.org/wiki/Hyperplane_separation_theorem
In geometry, the hyperplane separation theorem is a theorem about disjoint convex sets in n -dimensional Euclidean space. There are several rather similar versions. In one version of the theorem, if both these sets are closed and at least one of them is compact, then there is a …
https://math.stackexchange.com/questions/2097088/prove-the-supporting-hyperplane-theorem-for-convex-sets-in-euclidean-spaces
Just a little caveat: you might be very tempted to use, perhaps implicitly, some "advanced" properties of convex sets like "any convex set in $\Bbb R^n$ is the intersection of affine sets". But this may lead to circular reasoning, because, for instance, the proof of the property mentioned just above is, AFAIK, based exactly on the supporting ...
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