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High-dimensional limit theorems for random vectors in ℓpn-balls (2017)
Journal Article
Kabluchko, Z., Prochno, J., & Thäle, C. (2019). High-dimensional limit theorems for random vectors in ℓpn-balls. Communications in contemporary mathematics, 21(1), 1750092. doi:10.1142/S0219199717500924

In this paper, we prove a multivariate central limit theorem for ℓq-norms of high-dimensional random vectors that are chosen uniformly at random in an ℓnp-ball. As a consequence, we provide several applications on the intersections of ℓnp-balls in th... Read More about High-dimensional limit theorems for random vectors in ℓpn-balls.

On the expectation of operator norms of random matrices (2017)
Book Chapter
Guédon, O., Hinrichs, A., Litvak, A. E., & Prochno, J. (2017). On the expectation of operator norms of random matrices. In Lecture Notes in Mathematics; Geometric Aspects of Functional Analysis, 151-162. Springer International Publishing. doi:10.1007/978-3-319-45282-1_10

We prove estimates for the expected value of operator norms of Gaussian random matrices with independent (but not necessarily identically distributed) and centered entries, acting as operators from ℓnp∗ to ℓ q m , 1 ≤ p∗ ≤ 2 ≤ q  Read More about On the expectation of operator norms of random matrices.

On the geometry of projective tensor products (2017)
Journal Article
Giladi, O., Prochno, J., Schütt, C., Tomczak-Jaegermann, N., & Werner, E. (2017). On the geometry of projective tensor products. Journal of functional analysis, 273(2), 471-495. doi:10.1016/j.jfa.2017.03.019

© 2017 Elsevier Inc. In this work, we study the volume ratio of the projective tensor products ℓpn⊗πℓqn⊗πℓrnwith 1≤p≤q≤r≤∞. We obtain asymptotic formulas that are sharp in almost all cases. As a consequence of our estimates, these spaces allow for a... Read More about On the geometry of projective tensor products.

On the isotropic constant of random polytopes with vertices on an ℓp-Sphere (2017)
Journal Article
Hörrmann, J., Prochno, J., & Thäle, C. (2018). On the isotropic constant of random polytopes with vertices on an ℓp-Sphere. The Journal of geometric analysis, 28(1), 405-426. doi:10.1007/s12220-017-9826-z

The symmetric convex hull of random points that are independent and distributed according to the cone probability measure on the p-unit sphere of Rn for some 1 ≤ p < ∞ is considered. We prove that these random polytopes have uniformly absolutely boun... Read More about On the isotropic constant of random polytopes with vertices on an ℓp-Sphere.

On the geometry of random convex sets between polytopes and zonotopes (2017)
Journal Article
Alonso-Gutiérrez, D., & Prochno, J. (2017). On the geometry of random convex sets between polytopes and zonotopes. Journal of mathematical analysis and applications, 450(1), 670-690. doi:10.1016/j.jmaa.2017.01.042

In this work we study a class of random convex sets that "interpolate" between polytopes and zonotopes. These sets arise from considering a qth-moment (q≥1) of an average of order statistics of 1-dimensional marginals of a sequence of N≥n independent... Read More about On the geometry of random convex sets between polytopes and zonotopes.

Mean width of random perturbations of random polytopes (2017)
Journal Article
Alonso-Gutiérrez, D., & Prochno, J. (2017). Mean width of random perturbations of random polytopes. Advances in geometry, 17(1), 75-90. doi:10.1515/advgeom-2016-0032

We prove some “high probability” results on the expected value of the mean width for random perturbations of random polytopes. The random perturbations are considered for Gaussian random vectors and uniform distributions on ℓNp-balls and the unit sph... Read More about Mean width of random perturbations of random polytopes.

Estimating Averages of Order Statistics of Bivariate Functions (2016)
Journal Article
Lechner, R., Passenbrunner, M., & Prochno, J. (2017). Estimating Averages of Order Statistics of Bivariate Functions. Journal of theoretical probability, 30(4), 1445-1470. doi:10.1007/s10959-016-0702-8

© 2016, Springer Science+Business Media New York. We prove uniform estimates for the expected value of averages of order statistics of bivariate functions in terms of their largest values by a direct analysis. As an application, uniform estimates for... Read More about Estimating Averages of Order Statistics of Bivariate Functions.

Musielak-Orlicz spaces that are isomorphic to subspaces of L1 (2015)
Journal Article
Prochno, J. (2015). Musielak-Orlicz spaces that are isomorphic to subspaces of L1. Annals of Functional Analysis, 6(1), 84-94. doi:10.15352/afa/06-1-7

We prove that 1/n! ∑π∈Gn (∑ni=1 |xiai,π(i)|2)1/2 is equivalent to a Musielak-Orlicz norm ||x||∑Mi. We also provide the converse result, i.e., given the Orlicz functions, we provide a formula for the choice of the matrix that generates the correspondi... Read More about Musielak-Orlicz spaces that are isomorphic to subspaces of L1.

Probabilistic estimates for tensor products of random vectors (2015)
Journal Article
Alonso-Gutiérrez, D., Passenbrunner, M., & Prochno, J. (2016). Probabilistic estimates for tensor products of random vectors. Proceedings of the American Mathematical Society, 144(5), 2133-2148. doi:10.1090/proc/12883

We prove some probabilistic estimates for tensor products of random vectors, generalizing results that were obtained by Gordon, Litvak, Schütt,

Uniform estimates for averages of order statistics of matrices (2015)
Journal Article
Lechner, R., Passenbrunner, M., & Prochno, J. (2015). Uniform estimates for averages of order statistics of matrices. Electronic Communications in Probability, 20, 1-12. doi:10.1214/ECP.v20-3992

We prove uniform estimates for the expected value of averages of order statistics of matrices interms of their largest entries. As an application, we obtain similar probabilistic estimates for ℓp norms via real interpolation.

On the Gaussian behavior of marginals and the mean width of random polytopes (2015)
Journal Article
Alonso-Gutiérrez, D., & Prochno, J. (2015). On the Gaussian behavior of marginals and the mean width of random polytopes. Proceedings of the American Mathematical Society, 143(2), 821-832. https://doi.org/10.1090/S0002-9939-2014-12401-4

© 2014 American Mathematical Society. We show that the expected value of the mean width of a random polytope generated by N random vectors (n ≤ N ≤ e√n) uniformly distributed in an isotropic convex body in ℝnis of the order√logNLK. This completes a r... Read More about On the Gaussian behavior of marginals and the mean width of random polytopes.

On Mean Outer Radii of Random Polytopes (2014)
Journal Article
Alonso-Gutiérrez, D., Dafnis, N., Cifre, M. Á. H., & Prochno, J. (2014). On Mean Outer Radii of Random Polytopes. Indiana University Mathematics Journal, 63(2), 579-595

In this paper we introduce a new sequence of quantities for random polytopes. Let $K_N=\conv\{X_1,...,X_N\}$ be a random polytope generated by independent random vectors uniformly distributed in an isotropic convex body K of $\R^n$. We prove that the... Read More about On Mean Outer Radii of Random Polytopes.