Displaying similar documents to “A variational solution of the A. D. Aleksandrov problem of existence of a convex polytope with prescribed Gauss curvature”

Uniform decompositions of polytopes

Daniel Berend, Luba Bromberg (2006)

Applicationes Mathematicae

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We design a method of decomposing convex polytopes into simpler polytopes. This decomposition yields a way of calculating exactly the volume of the polytope, or, more generally, multiple integrals over the polytope, which is equivalent to the way suggested in Schechter, based on Fourier-Motzkin elimination (Schrijver). Our method is applicable for finding uniform decompositions of certain natural families of polytopes. Moreover, this allows us to find algorithmically an analytic expression...

Approximation of the Euclidean ball by polytopes

Monika Ludwig, Carsten Schütt, Elisabeth Werner (2006)

Studia Mathematica

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There is a constant c such that for every n ∈ ℕ, there is an Nₙ so that for every N≥ Nₙ there is a polytope P in ℝⁿ with N vertices and v o l ₙ ( B ₂ ⁿ △ P ) ≤ c v o l ₙ ( B ₂ ⁿ ) N - 2 / ( n - 1 ) where B₂ⁿ denotes the Euclidean unit ball of dimension n.

An inequality concerning edges of minor weight in convex 3-polytopes

Igor Fabrici, Stanislav Jendrol' (1996)

Discussiones Mathematicae Graph Theory

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Let e i j be the number of edges in a convex 3-polytope joining the vertices of degree i with the vertices of degree j. We prove that for every convex 3-polytope there is 20 e 3 , 3 + 25 e 3 , 4 + 16 e 3 , 5 + 10 e 3 , 6 + 6 [ 2 / 3 ] e 3 , 7 + 5 e 3 , 8 + 2 [ 1 / 2 ] e 3 , 9 + 2 e 3 , 10 + 16 [ 2 / 3 ] e 4 , 4 + 11 e 4 , 5 + 5 e 4 , 6 + 1 [ 2 / 3 ] e 4 , 7 + 5 [ 1 / 3 ] e 5 , 5 + 2 e 5 , 6 ≥ 120 ; moreover, each coefficient is the best possible. This result brings a final answer to the conjecture raised by B. Grünbaum in 1973.

Coating by cubes.

Bezdek, K., Hausel, T. (1994)

Beiträge zur Algebra und Geometrie

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Rigidity and flexibility of virtual polytopes

G. Panina (2003)

Open Mathematics

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All 3-dimensional convex polytopes are known to be rigid. Still their Minkowski differences (virtual polytopes) can be flexible with any finite freedom degree. We derive some sufficient rigidity conditions for virtual polytopes and present some examples of flexible ones. For example, Bricard's first and second flexible octahedra can be supplied by the structure of a virtual polytope.