## Annals of Functional Analysis

### Polytopes of stochastic tensors

#### Abstract

Considering $n\times n\times n$ stochastic tensors $(a_{ijk})$ (i.e., nonnegative hypermatrices in which every sum over one index $i$, $j$, or $k$, is 1), we study the polytope ($\Omega_{n}$) of all these tensors, the convex set ($L_{n}$) of all tensors in $\Omega_{n}$ with some positive diagonals, and the polytope ($\Delta_{n}$) generated by the permutation tensors. We show that $L_{n}$ is almost the same as $\Omega_{n}$ except for some boundary points. We also present an upper bound for the number of vertices of $\Omega_{n}$.

#### Article information

Source
Ann. Funct. Anal., Volume 7, Number 3 (2016), 386-393.

Dates
Accepted: 26 November 2015
First available in Project Euclid: 19 May 2016

https://projecteuclid.org/euclid.afa/1463684084

Digital Object Identifier
doi:10.1215/20088752-3605195

Mathematical Reviews number (MathSciNet)
MR3506610

Zentralblatt MATH identifier
1347.15040

Subjects
Primary: 15B51: Stochastic matrices
Secondary: 52B11: $n$-dimensional polytopes

#### Citation

Chang, Haixia; Paksoy, Vehbi E.; Zhang, Fuzhen. Polytopes of stochastic tensors. Ann. Funct. Anal. 7 (2016), no. 3, 386--393. doi:10.1215/20088752-3605195. https://projecteuclid.org/euclid.afa/1463684084

#### References

• [1] M. Ahmed, J. De Loera, and R. Hemmecke, “Polyhedral cones of magic cubes and squares” in Discrete and Computational Geometry, Algorithms Combin. 25, Springer, Berlin, 2003, 25–41.
• [2] A. Brondsted, An Introduction to Convex Polytopes, Grad. Texts in Math. 90, Springer, New York, 1983.
• [3] R. A. Brualdi and J. Csima, Stochastic patterns, J. Combin. Theory Ser. A 19 (1975) 1–12.
• [4] K. Chang, L. Qi, and T. Zhang, A survey on the spectral theory of nonnegative tensors, Numer. Linear Algebra Appl. 20 (2013), no. 6, 891–912.
• [5] J. Csima, Multidimensional stochastic matrices and patterns, J. Algebra 14 (1970), 194–202.
• [6] L.-B. Cui, W. Li, and M. K. Ng, Birkhoff–von Neumann theorem for multistochastic tensors, SIAM. J. Matrix Anal. & Appl. 35 (2014), no. 3, 956–973.
• [7] P. Fischer and E. R. Swart, Three dimensional line stochastic matrices and extreme points, Linear Algebra Appl. 69 (1985), 179–203.
• [8] R. A. Horn and C. R. Johnson, Matrix Analysis, 2nd ed., Cambridge Univ. Press, Cambridge, 2013.
• [9] T. G. Kolda and B. W. Bader, Tensor Decompositions and Applications, SIAM Rev. 51 (2009), no. 3, 455–500.
• [10] L.-H. Lim, “Tensors and hypermatrices” in Handbook of Linear Algebra, 2nd ed., Discrete Math. Appl. (Boca Raton), Chapman and Hall/CRC, Boca Raton, FL, 2013, Chapter 15.
• [11] G. M. Ziegler, Lectures on Polytopes, Grad. Texts in Math. 152, Springer, New York, 1995.