Geometry of Log-concave Functions and Measures

By Bo'az Klartag, V. Milman. 2005

170 citations recorded by the source corpus.

Subfields: Applied Mathematics, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/028db89cacbd5ae3dcd017d7fb617edffe02985a

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In Applied Mathematics

In Molecular Biology

Citations in the corpus

Hidden structures in the class of convex functions and a new duality transform (2011, background): “(We remark that some such extensions for operations involving L were done earlier: see [2], [13] and [12]; in particular, the Santaló inequality was proved in [2] for the duality involving L.”
Some functional forms of Blaschke–Santaló inequality (2006, methodology): “Notice that Lutwak and Zhang [15] and Lutwak, Yang and Zhang [14] gave other very different functional forms of the Blaschke-Santaló inequality and recently Klartag and Milman [13], Klartag [12] and Colesanti [6] also established functional forms of some other geometric inequalities.”
Valuations on Convex Functions (2017, background): “Moreover, classical functionals for convex sets including the intrinsic volumes have been extended to the space of quasi-concave functions in [7] and [28] (see also [9, 18]).”
Uniform almost sub-Gaussian estimates for linear functionals on convex sets (2007, methodology): “The next lemma is, again, very similar to the methods in [21].”
Increasing functions and inverse Santaló inequality for unconditional functions (2008, methodology): “This statement was proved for even functions by K. Ball [2] and in full generality by Artstein-Klartag-Milman [1] (see also Fradelizi-Meyer [8]).”
Concentration of the information in data with log-concave distributions (2010, methodology): “A similar concentration inequality was obtained by Klartag and Mil-man [15], who compared the value f ( X ) to the maximum M of the density f and proved that with some absolute constants c 0 , c 1 ∈ (0 , 1).”
Rogers–Shephard inequality for log-concave functions (2014, background): “In [18,22,30], the case of the perimeter and the mean width is considered while, in [13], a different definition is given for all the quermaßintegrals.”
On the infimum convolution inequality (2008, methodology): “been previously used in the context of convex geometry (see for instance [2] and [15]).”
Quermassintegrals of quasi-concave functions and generalized Prékopa–Leindler inequalities (2012, methodology): “To that aim, we exploit as a crucial tool the concept of the functional projection introduced in [22].”
A central limit theorem for convex sets (2006, background): “This concentration of measure phenomenon is one of the main driving forces in high-dimensional convex geometry, as was first demonstrated by Milman in his proof of Dvoretzky’s theorem (see [32] or [15, Section 4.2]).”
Functional Versions of Lp-Affine Surface Area and Entropy Inequalities (2014, unknown): “For all λ > 1, we go back to Corollary 1 and deduce that By the asymptotic functional reverse Santal´o inequality [21] (see also [27] in the even case), there exists a constant c > 0 such that R n e − ψ R n e − ψ ∗ ≥ c n .”
A characterization of the concept of duality (2007, unknown): “It is defined (again, after fixing a scalar product on Rn) for a function f = e−φ by the formula f◦ = e−Lφ, or, more explicitly, by (2) f◦(x) = inf y∈Rn e−〈x,y〉 f(y) .”
Reverse Brunn–Minkowski and reverse entropy power inequalities for convex measures (2011, background): “Milman in [30], who proved the following statement (cf.”
Some functional inverse Santalo inequalities (2008, background): “Klartag and Milman [10] proved that there exists a constant c > 0 such that for every n ! 1 and every logconcave function f : Rn → R+ satisfying 0 < ∫ Rn f (x) dx < +∞ and the additional hypothesis f (0) = maxf , then ∫”