Abstract
In this paper, we deepen the theoretical study of the geometric structure of a balanced complex polytope (b.c.p.), which is the generalization of a real centrally symmetric polytope to the complex space. We also propose a constructive algorithm for the representation of its facets in terms of their associated linear functionals. The b.c.p.s are used, for example, as a tool for the computation of the joint spectral radius of families of matrices. For the representation of real polytopes, there exist well-known algorithms such as, for example, the Beneath–Beyond method. Our purpose is to modify and adapt this method to the complex case by exploiting the geometric features of the b.c.p. However, due to the significant increase in the difficulty of the problem when passing from the real to the complex case, in this paper, we confine ourselves to examine the two-dimensional case. We also propose an algorithm for the computation of the norm the unit ball of which is a b.c.p.
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References
R.K. Brayton, C.H. Tong, Constructive stability and asymptotic stability of dynamical systems, IEEE Trans. Circuits Syst. 27, 1121–1130 (1980).
H. Edelsbrunner, Algorithms in Combinatorial Geometry, EATCS Monographs on Theoretical Computer Science (Springer, Heidelberg, 1987).
B. Grünbaum, Convex Polytopes (Wiley, London, 1967).
N. Guglielmi, F. Wirth, M. Zennaro, Complex polytope extremality results for families of matrices, SIAM J. Matrix Anal. Appl. 27, 721–743 (2005).
N. Guglielmi, M. Zennaro, Balanced complex polytopes and related vector and matrix norms, J. Convex Anal. 14, 729–766 (2007).
N. Guglielmi, M. Zennaro, An algorithm for finding extremal polytope norms of matrix families, Linear Algebra Appl. 428, 2265–2282 (2008).
M. Maesumi, Construction of optimal norms for semigroups of matrices, in 44th IEEE Conference on Decision and Control and European Control Conference ECC’05, pp. 3001–3006, Seville, Spain, 12–15 December 2005.
S. Miani, C. Savorgnan, Complex polytopic control Lyapunov functions, in 45th IEEE Conference on Decision and Control, pp. 3198–3203, San Diego, CA, USA, 13–15 December 2006.
C. Vagnoni, Algorithms for the computation of the joint spectral radius, PhD Thesis, The University of Padova, 2008.
G.M. Ziegler, Lectures on Polytopes (Springer, New York, 1995).
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Communicated by Herbert Edelsbrunner.
This work was supported by INdAM-GNCS.
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Vagnoni, C., Zennaro, M. The Analysis and the Representation of Balanced Complex Polytopes in 2D. Found Comput Math 9, 259–294 (2009). https://doi.org/10.1007/s10208-008-9032-2
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DOI: https://doi.org/10.1007/s10208-008-9032-2