Abstract
Isotropy-like properties are considered for finite measures with heavy tails. As a basic tool, we extend K. Ball’s relationship between convex bodies and finite logarithmically concave measures to a larger class of distributions, satisfying convexity conditions of the Brunn–Minkowski type.
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Artstein, S., Klartag, B., Milman, V.: The Santaló point of a function, and a functional form of the Santaló inequality. Mathematika, 51(1–2) (2004), 33–48 (2005)
Artstein, S., Milman, V.: The concept of duality in asymptotic geometric analysis, and the characterization of the Legendre transform. Preprint (2007)
Artstein S., Milman V.: A characterization of the concept of duality. Electron. Res. Announc. Math. Sci. 14, 42–59 (2007) (electronic)
Artstein S., Milman V.: The concept of duality for measure projections of convex bodies. J. Funct. Anal. 254(10), 2648–2666 (2008)
Avriel M.: r-convex functions. Math. Programming 2, 309–323 (1972)
Ball, K.M.: Isometric Problems in ℓ p and Sections of Convex Sets. PhD Dissertation, Cambridge (1986)
Ball K.M.: Logarithmically concave functions and sections of convex bodies. Studia Math. 88, 69–84 (1988)
Birch B.J.: On 3N points in a plane. Proc. Cambridge Philos. Soc. 55, 289–293 (1959)
Bobkov S.G.: Isoperimetric and analytic inequalities for log-concave probability measures. Ann. Probab. 27(4), 1903–1921 (1999)
Bobkov S.G.: On concentration of distributions of random weighted sums. Ann. Probab. 31(1), 195–215 (2003)
Bobkov S.G.: Large deviations and isoperimetry over convex probability measures with heavy tails. Electron. J. Probab. 12, 1072–1100 (2007)
Borell C.: Convex measures on locally convex spaces. Ark. Math. 12, 239–252 (1974)
Borell C.: Convex set functions in d-space. Period. Math. Hungar. 6(2), 111–136 (1975)
Bourgain J.: On high-dimensional maximal functions associated to convex bodies. Am. J. Math. 108(6), 1467–1476 (1986)
Bourgain, J.: On the distribution of polynomials on high-dimensional convex sets. Geometric Aspects of Functional Analysis (1989–1990), Lecture Notes in Mathematics, vol. 1469, pp. 127–137. Springer, Berlin (1991)
Brascamp H.J., Lieb E.H.: On extensions of the Brunn–Minkowski and Pre’kopa–Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation. J. Funct. Anal. 22(4), 366–389 (1976)
Caplin A., Nalebuff B.: Aggregation and social choice: a mean voter theorem. Econometrica 59(1), 1–23 (1991)
Das Gupta S.: Brunn–Minkowski inequality and its aftermath. J. Multivar. Anal. 10(3), 296–318 (1980)
Dancs S., Uhrin B.: On a class of integral inequalities and their measure-theoretic consequences. J. Math. Anal. Appl. 74(2), 388–400 (1980)
Fradelizi M.: Hyperplane sections of convex bodies in isotropic position. Beiträge Algebra Geom. 40(1), 163–183 (1999)
Fradelizi M., Meyer M.: Some functional forms of Blaschke-Santaló inequality. Math. Z. 256(2), 379–395 (2007)
Gardner, R.J.: Geometric Tomography. Encyclopedia of Mathematics and its Applications, vol. 58, xvi+424 pp. Cambridge University Press, Cambridge (1995)
Giannopoulos A.A., Milman V.D.: Extremal problems and isotropic positions of convex bodies. Israel J. Math. 117, 29–60 (2000)
Gorin, E.A., Favorov, S. Yu.: Generalizations of the Khinchin inequality. (Russian) Teor. Veroyatnost. i Primenen. 35(4), 762–767 (1990); translation in Theory Probab. Appl., 35(4), 766–771 (1990)
Guédon O.: Kahane–Khinchine type inequalities for negative exponent. Mathematika 46(1), 165–173 (1999)
Grünbaum B.: Partitions of mass-distributions and of convex bodies by hyperplanes. Pac. J. Math. 10, 1257–1261 (1960)
Hammer, P.C.: Volumes Cut from Convex Bodies by Planes. Preprint (this reference is given in [26])
Hensley D.: Slicing convex bodies—bounds for slice area in terms of the body’s covariance. Proc. Am. Math. Soc. 79(4), 619–625 (1980)
Koldobsky, A.L.: Fourier analysis in convex geometry. Mathematical Surveys and Monographs, vol. 116, vi+170 pp. American Mathematical Society, Providence (2005)
Leichtweiss K.: On the affine surface of convex bodies (in German). Manuscripta Math. 56(4), 429–464 (1986)
Leichtweiss K.: On a formula of Blaschke on the affine surface area (in German). Studia Sci. Math. Hungar. 21(3–4), 453–474 (1986)
Meyer M., Reisner S.: A geometric property of the boundary of symmetric convex bodies and convexity of flotation surfaces. Geom. Dedicata 37(3), 327–337 (1991)
Meyer, M., Reisner, S.: Characterizations of affinely-rotation-invariant log-concave measures by section-centroid location. Geom. Aspects of Funct. Anal. (1989–1990), Lecture Notes in Mathematics, vol. 1469, pp. 145–152. Springer, Berlin (1991)
Milman, V.D., Pajor, A.: Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space. Geom. Aspects of Funct. Anal. (1987–1988), Lecture Notes in Mathematics, vol. 1376, pp. 64–104. Springer, Berlin (1989)
Neumann B.H.: On an invariant of plane regions and mass distributions. J. Lond. Math. Soc. 20, 226–237 (1945)
Pisier G.: The Volume of Convex Bodies and Banach Space Geometry. Cambridge University Press, Cambridge (1989)
Santaló L.A.: An affine invariant for convex bodies of n-dimensional space (Spanish). Port. Math. 8, 155–161 (1949)
Rado R.: A theorem on general measure. J. Lond. Math. Soc. 21, 291–300 (1946)
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Supported in part by the NSF grant DMS-0706866.
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Bobkov, S.G. Convex bodies and norms associated to convex measures. Probab. Theory Relat. Fields 147, 303–332 (2010). https://doi.org/10.1007/s00440-009-0209-7
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DOI: https://doi.org/10.1007/s00440-009-0209-7