Penumbras and Separation of Convex Sets

Soltan, Valeriu

doi:10.1007/s00025-020-01329-7

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Penumbras and Separation2

Given Convex Sets K1

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2021

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“…17). It is possible to show (see [80]) that both sets P (K 1 , K 2 ) and P (K 2 , K 1 ) are convex and contain K 1 and K 2 , respectively. The following theorem illustrates the role of penumbras in separation of convex sets.…”

Section: Penumbras and Separationmentioning

confidence: 99%

“…H Theorem 5.12. ( [80]) Let K 1 and K 2 be convex sets in R n . A hyperplane H ⊂ R n separates (respectively, properly, strictly, or strongly) K 1 and K 2 if and only if it separates (respectively, nontrivially, strictly, or strongly) the sets P (K 1 , K 2 ) and P (K 2 , K 1 ).…”

Section: Penumbras and Separationmentioning

confidence: 99%

“…The following assertions from [80] relate various properties of penumbras to some known classes of convex sets in R n .…”

Section: Given Convex Sets Kmentioning

confidence: 99%

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Support and separation properties of convex sets in finite dimension

Soltan

2021

Extr. Math.

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This is a survey on support and separation properties of convex sets in the n-dimensional Euclidean space. It contains a detailed account of existing results, given either chronologically or in related groups, and exhibits them in a uniform way, including terminology and notation. We first discuss classical Minkowski’s theorems on support and separation of convex bodies, and next describe various generalizations of these results to the case of arbitrary convex sets, which concern bounding and asymptotic hyperplanes, and various types of separation by hyperplanes, slabs, and complementary convex sets.

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Section: Penumbras and Separationmentioning

confidence: 99%

Section: Penumbras and Separationmentioning

confidence: 99%