منابع مشابه
Approximately Gaussian marginals and the hyperplane conjecture
We discuss connections between certain well-known open problems related to the uniform measure on a high-dimensional convex body. In particular, we show that the “thin shell conjecture” implies the “hyperplane conjecture”. This extends a result by K. Ball, according to which the stronger “spectral gap conjecture” implies the “hyperplane conjecture”.
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The Monodromy Conjecture asserts that if c is a pole of the local topological zeta function of a hypersurface, then exp(2πic) is an eigenvalue of the monodromy on the cohomology of the Milnor fiber. A stronger version of the conjecture asserts that every such c is a root of the Bernstein-Sato polynomial of the hypersurface. In this note we prove the weak version of the conjecture for hyperplane...
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We prove that a real analytic subset of a torus group that is contained in its image under an expanding endomorphism is a finite union of translates of closed subgroups. This confirms the hyperplane zeros conjecture of Lagarias and Wang for real analytic varieties. Our proof uses real analytic geometry, topological dynamics, and Fourier analysis.
متن کاملTHE VARIANCE CONJECTURE ON HYPERPLANE PROJECTIONS OF THE `p BALLS
We show that for any 1 ≤ p ≤ ∞, the family of random vectors uniformly distributed on hyperplane projections of the unit ball of `p verify the variance conjecture Var |X| ≤ C max ξ∈Sn−1 E〈X, ξ〉E|X|, where C depends on p but not on the dimension n or the hyperplane. We will also show a general result relating the variance conjecture for a random vector uniformly distributed on an isotropic conve...
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This seemingly innocuous question, considered two decades ago by Bourgain [3, 4], has not been answered yet. We refer the reader to, e.g., [2], [18] or [14] for partial results, history and for additional literature regarding this hyperplane conjecture. In particular, there are large classes of convex bodies for which an affirmative answer to the above question is known. These include unconditi...
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ژورنال
عنوان ژورنال: Acta Mathematica Sinica, English Series
سال: 2019
ISSN: 1439-8516,1439-7617
DOI: 10.1007/s10114-019-8241-8