## Electronic Communications in Probability

### The monotonicity of $f$-vectors of random polytopes

#### Abstract

Let $K$ be a compact convex body in ${\mathbb R}^d$, let $K_n$ be the convex hull of $n$ points chosen uniformly and independently in $K$, and let $f_{i}(K_n)$ denote the number of $i$-dimensional faces of $K_n$. ;We show that for planar convex sets, $E[f_0 (K_n)]$ is increasing in $n$.  In dimension $d \geq 3$ we prove that if $\lim_{n \to \infty} \frac{E[f_{d-1}(K_n)]}{An^c}=1$ for some constants $A$ and $c>0$ then the function $n \mapsto E[f_{d-1}(K_n)]$ is increasing for $n$ large enough. In particular, the number of facets of the convex hull of $n$ random points distributed uniformly and independently in a smooth compact convex body is asymptotically increasing. Our proof relies on a random sampling argument.

#### Article information

Source
Electron. Commun. Probab., Volume 18 (2013), paper no. 23, 8 pp.

Dates
Accepted: 28 March 2013
First available in Project Euclid: 7 June 2016

https://projecteuclid.org/euclid.ecp/1465315562

Digital Object Identifier
doi:10.1214/ECP.v18-2469

Mathematical Reviews number (MathSciNet)
MR3044471

Zentralblatt MATH identifier
1359.60024

Rights

#### Citation

Devillers, Olivier; Glisse, Marc; Goaoc, Xavier; Moroz, Guillaume; Reitzner, Matthias. The monotonicity of $f$-vectors of random polytopes. Electron. Commun. Probab. 18 (2013), paper no. 23, 8 pp. doi:10.1214/ECP.v18-2469. https://projecteuclid.org/euclid.ecp/1465315562

#### References

• BÃ¡rÃ¡ny, I.; Larman, D. G. Convex bodies, economic cap coverings, random polytopes. Mathematika 35 (1988), no. 2, 274–291.
• Buchta, Christian; Reitzner, Matthias. Equiaffine inner parallel curves of a plane convex body and the convex hulls of randomly chosen points. Probab. Theory Related Fields 108 (1997), no. 3, 385–415.
• Chazelle, Bernard. The discrepancy method. Randomness and complexity. Cambridge University Press, Cambridge, 2000. xviii+463 pp. ISBN: 0-521-77093-9
• Clarkson, Kenneth L. New applications of random sampling in computational geometry. Discrete Comput. Geom. 2 (1987), no. 2, 195–222.
• Clarkson, Kenneth L.; Shor, Peter W. Applications of random sampling in computational geometry. II. Discrete Comput. Geom. 4 (1989), no. 5, 387–421.
• Dafnis, N.; Giannopoulos, A.; Guédon, O. On the isotropic constant of random polytopes. Adv. Geom. 10 (2010), no. 2, 311–322.
• Edelsbrunner, Herbert. Algorithms in combinatorial geometry. EATCS Monographs on Theoretical Computer Science, 10. Springer-Verlag, Berlin, 1987. xvi+423 pp. ISBN: 3-540-13722-X
• Efron, Bradley. The convex hull of a random set of points. Biometrika 52 1965 331–343.
• M. Meckes. Monotonicity of volumes of random simplices In: Recent Trends in Convex and Discrete Geometry. AMS Special Session 2006, San Antonio, Texas http://math.gmu.edu/simvsoltan/SanAntonio_06.pdf
• Milman, V. D.; Pajor, A. Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed $n$-dimensional space. Geometric aspects of functional analysis (1987â€“88), 64–104, Lecture Notes in Math., 1376, Springer, Berlin, 1989.
• Preparata, Franco P.; Shamos, Michael Ian. Computational geometry. An introduction. Texts and Monographs in Computer Science. Springer-Verlag, New York, 1985. xii+390 pp. ISBN: 0-387-96131-3
• Rademacher, Luis. On the monotonicity of the expected volume of a random simplex. Mathematika 58 (2012), no. 1, 77–91.
• Reitzner, Matthias. The combinatorial structure of random polytopes. Adv. Math. 191 (2005), no. 1, 178–208.
• Reitzner, Matthias. Random polytopes. New perspectives in stochastic geometry, 45–76, Oxford Univ. Press, Oxford, 2010.
• Schneider, Rolf; Weil, Wolfgang. Stochastic and integral geometry. Probability and its Applications (New York). Springer-Verlag, Berlin, 2008. xii+693 pp. ISBN: 978-3-540-78858-4
• Vu, V. H. Sharp concentration of random polytopes. Geom. Funct. Anal. 15 (2005), no. 6, 1284–1318.