Affine isoperimetric inequalities compare functionals, associated with convex (or more general) bodies, whose ratios are invariant under GL(n)-transformations of the bodies. These isoperimetric inequalities are more powerful than their better-known relatives of a Euclidean flavor. To be a bit more specific, this article deals with inequalities for centroid and projection bodies. Centroid bodies...

We prove several results of the following type: given finite dimensional normed space V there exists another space X with log dimX = O(log dimV ) and such that every subspace (or quotient) of X, whose dimension is not “too small,” contains a further subspace isometric to V . This sheds new light on the structure of such large subspaces or quotients (resp., large sections or projections of conve...

We verify a conjecture of Lutwak, Yang, Zhang about the equality case in the Orlicz-Petty projection inequality, and provide an essentially optimal stability version. The Petty projection inequality (Theorem 1), its Lp extension, and its analytic counterparts, the Zhang-Sobolev inequality [43] and its Lp extension by A. Cianchi, E. Lutwak, D. Yang, G. Zhang [8, 32], are fundamental affine isope...

Many crucial results of the asymptotic theory of symmetric convex bodies were extended to the non-symmetric case in recent years. That led to the conjecture that for every n-dimensional convex body K there exists a projection P of rank k, proportional to n, such that PK is almost symmetric. We prove that the conjecture does not hold. More precisely, we construct an n-dimensional convex body K s...

We formulate the many-body spin dynamics at high temperature within the non-equilibrium Keldysh formalism. For the simplest XY interaction, analytical expressions in terms of the one particle solutions are obtained for linear and ring configurations. For small rings of even spin number, the group velocities of excitations depend on the parity of the total spin projection. This should enable a d...

We provide an affirmative answer to a problem posed by Barvinok and Veomett in [4], showing that in general an n-dimensional convex body cannot be approximated by a projection of a section of a simplex of sub-exponential dimension. Moreover, we prove that for all 1 ≤ n ≤ N there exists an n-dimensional convex body B such that for every n-dimensional convex body K obtained as a projection of a s...

Information received by the human cortex is supplied by two main sources: extrinsic stimuli delivered by the external environment and intrinsic information regarding the body and self. We reanalyzed electrophysiological data involving the same external stimuli, but manipulating the degree of 'self-projection' to locations inside and outside the body border. Electrical neuroimaging and spatial p...

(a) Ground Truth (b) Our method (c) Ground Truth (d) Our method Figure 1. 3D human pose estimation on Human3.6m for several different action categories (a,c) Ground-truth 3D poses and their projection in the images. (b,d) The skeletons recovered by our method and their projection on the image plane. We can reliably recover the 3D pose of the body in case of ambiguities, such as self-occlusions ...