Around A. D. Alexandrov’s uniqueness theorem for convex polytopes

Panina, Gaiane

doi:10.1515/advgeom-2012-0006

Cited by 4 publications

(6 citation statements)

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“…Теорема единственности А. Д. Александрова для выпуклых многогранников и ее уточнения. Иное применение гиперболических многогранников можно найти в [31]. Оно непосредственно связано со следующей теоремой, которую А. Д. Александров расценивал как дискретный вариант теоремы о единственности выпуклых поверхностей [2].…”

Section: г ю панина и стрейнуunclassified

“…Теорема 28 [31]. Существуют два различных трехмерных выпуклых многогранника, для каждой пары параллельных гиперграней которых найдется не более одного параллельного переноса, помещающего одну грань строго в другую.…”

Section: г ю панина и стрейнуunclassified

“…Теорема 29 [31]. Пусть K , M -выпуклые трехмерные многогранники, для каждой пары параллельных гиперграней которых верно следующее:…”

Section: г ю панина и стрейнуunclassified

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Virtual polytopes

Panina¹,

Panina²,

Streinu³

2015

Uspekhi Mat. Nauk

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Появившиеся в различных областях математики-от алгебры многогранников и торических многообразий до теории вложенных графов с внутренними напряжениями, виртуальные многогранники представляют собой естественное алгебраическое обобщение выпуклых многогранников. Будучи определенными как элементы группы Гротендика, связанной с полугруппой выпуклых многогранников, они, однако, могут быть представлены геометрически. Данный обзор связывает теорию виртуальных многогранников с другими областями геометрии, содержит описание разнообразных геометризаций с взаимными связями, а также дает ряд приложений этой теории. Библиография: 50 названий.

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Section: г ю панина и стрейнуunclassified

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Virtual polytopes

Panina¹,

Panina²,

Streinu³

2015

Uspekhi Mat. Nauk

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“…Recall that the support function of a convex polytope is convex, i.e. its graph is a convex surface (it is reasonable to consider either the spherical graph or the collection of affine graphs [18], [19]).…”

Section: Hyperbolic Virtual Polytopesmentioning

confidence: 99%

“…(1) A refinement of A.D. Alexadrov's unieqness theorem for 3D-polytopes with non-insertable pairs of parallel faces [19]. (2) Extrinsic geometry of saddle surfaces with injective Gaussian mapping [18][19][20][21]. (3) Isotopy problem for saddle surfaces [21].…”

Section: Lemma 22 (See [20])mentioning

confidence: 99%

Pointed spherical tilings and hyperbolic virtual polytopes

Panina

2011

J Math Sci

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We wish to draw attention to an interesting and promising interaction of two theories.On the one hand, it is the theory of pseudo-triangulations which was useful for implicit solution of the carpenter's rule problem and proved later to give a nice tool for graph embeddings.On the other hand, it is the theory of hyperbolic virtual polytopes which arose from an old uniqueness conjecture for convex bodies (A. D. Alexandrov's problem): suppose that a constant C separates (non-strictly) everywhere the principal curvature radii of a smooth 3-dimensional convex body K. Then K is necessarily a ball of radius C.The two key ideas are:• Passing from planar pseudo-triangulations to spherical pseudo-tilings, we avoid non-poited vertices. Instead, we use pseudo-di-gons. A theorem on spherically embedded Laman-plus-one graphs is announced. • The difficult problem of hyperbolic polytopes constructing can be reduced to finding spherically embedded graphs.

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Why there is no an existence theorem for a convex polytope with prescribed directions and perimeters of the faces?

Alexandrov

2017

Abh. Math. Semin. Univ. Hambg.

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We choose some special unit vectors n 1 , . . . , n 5 in R 3 and denote by L ⊂ R 5 the set of all points (L 1 , . . . , L 5 ) ∈ R 5 with the following property: there exists a compact convex polytope P ⊂ R 3 such that the vectors n 1 , . . . , n 5 (and no other vector) are unit outward normals to the faces of P and the perimeter of the face with the outward normal n k is equal to L k for all k = 1, . . . , 5. Our main result reads that L is not a locally-analytic set, i. e., we prove that, for some point (L 1 , . . . , L 5 ) ∈ L, it is not possible to find a neighborhood U ⊂ R 5 and an analytic set A ⊂ R 5 such that L ∩ U = A ∩ U . We interpret this result as an obstacle for finding an existence theorem for a compact convex polytope with prescribed directions and perimeters of the faces. Mathematics Subject Classification (2010): 52B10; 51M20.

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Around A. D. Alexandrov’s uniqueness theorem for convex polytopes

Cited by 4 publications

References 6 publications

Virtual polytopes

Virtual polytopes

Pointed spherical tilings and hyperbolic virtual polytopes

Why there is no an existence theorem for a convex polytope with prescribed directions and perimeters of the faces?

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