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- J.-P. Bouchaud, M. Potters, M. Meyer
- 1999

We present a exactly soluble model for financial time series that mimics the long range volatility correlations known to be present in financial data. Although our model is asymptotically 'monofractal' by construction, it shows apparent multiscaling as a result of a slow crossover phenomenon on finite time scales. Our results suggest that it might be hard… (More)

We prove that the reciprocal of the volume of the polar bodies, about the Santaló point, of a shadow system of convex bodies K t , is a convex function of t. Thus extending to the non-symmetric case a result of Campi and Gronchi. The case that the reciprocal of the volume is an affine function of t is also investigated and is characterized under certain… (More)

We answer in the negative a question by Grünbaum who asked if there exists a finite basis of affine invariant points. We give a positive answer to another question by Grünbaum about the " size " of the set of all affine invariant points. Related, we show that the set of all convex bodies K, for which the set of affine invariant points is all of R n , is… (More)

Grünbaum introduced measures of symmetry for convex bodies that measure how far a given convex body is from a centrally symmetric one. Here, we introduce new measures of symmetry that measure how far a given convex body is from one with " enough symmetries ". To define these new measures of symmetry, we use affine covariant points. We give examples of… (More)

An affine invariant point on the class of convex bodies Kn in R n , endowed with the Hausdorff metric, is a continuous map from Kn to R n which is invariant under one-to-one affine transformations A on R n , that is, p ` A(K) ´ = A ` p(K) ´. We define here the new notion of dual affine point q of an affine invariant point p by the formula q(K p(K)) = p(K)… (More)

We elaborate on the use of shadow systems to prove a particular case of the conjectured lower bound of the volume product P(K) = min z∈int(K) |K|||K z |, where K ⊂ R n is a convex body and K z = {y ∈ R n : (y − z) · (x − z) 1 for all x ∈ K} is the polar body of K with respect to the center of polarity z. In particular, we show that if K ⊂ R 3 is the convex… (More)

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