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We provide a generalization of John's representation of the identity for the maximal volume position of L inside K, where K and L are arbitrary smooth convex bodies in R n. >From this representation we obtain Banach-Mazur distance and volume ratio estimates.

- N Dafnis, A Giannopoulos, A Tsolomitis
- 2012

Let K be an isotropic convex body in R xN are independent random points, uniformly distributed in K. We prove that if n 2 N exp(√ n) then the normalized quermaßintegrals

Let K be a convex body in R n and let be its quermassintegrals. We study minimization problems of the form minfW i (TK) j T 2 SL n g and show that bodies which appear as solutions of such problems satisfy isotropic conditions or even admit an isotropic characterization for appropriate measures. This shows that several well known positions of convex bodies… (More)

- N Dafnis, A Giannopoulos, O Guédon
- 2008

Let K be an isotropic 1-unconditional convex body in R n. For every N > n consider N independent random points x1,. .. , xN uniformly distributed in K. We prove that, with probability greater than 1 − C 1 exp(−cn) if N ≥ c 1 n and greater than 1−C 1 exp(−cn/ log n) if n < N < c 1 n, the random

- A Giannopoulos, A Pajor, G Paouris
- 2006

We give an alternative proof of a recent result of Klartag on the existence of almost subgaussian linear functionals on convex bodies. If K is a convex body in R n with volume one and center of mass at the origin, there exists x = 0 such that |{y ∈ K : |y, x| t·, x1}| exp(−ct 2 / log 2 (t + 1)) for all t 1, where c > 0 is an absolute constant. The proof is… (More)

- A A Giannopoulos, V D Milman, M K D N ∩ F
- 1996

Let K be a symmetric convex body in R n. It is well-known that for every θ ∈ (0, 1) there exists a subspace F of R n with dim F = [(1 − θ)n] such that where P F denotes the orthogonal projection onto F. Consider a fixed coordinate system in R n. We study the question whether an analogue of (*) can be obtained when one is restricted to choose F among the… (More)