After setting geometric notions, we revisit an exponential functional which has arisen in several contexts, with special attention to a set of geometric parameters and associated inequalities.
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C. Borell, “On a certain exponential inequality for Gaussian processes,” Extremes, 9, 169–176 (2006).
S. Chevet, “Processus gaussiens et volumes mixtes,” Z. Wahrsch. Verw. Gebiete, 36, 47–65 (1976).
R. M. Dudley, “The sizes of compact subsets of Hilbert space and continuity of Gaussian processes,” J. Functional Analysis, 1, 290–330 (1967).
X. Fernique, “Corps convexes et processus gaussiens de petit rang,” Z. Wahrsch. Verw. Gebiete, 35, 349–353 (1976).
L. Gurvits, “A short proof, based on mixed volumes, of Liggett’s theorem on the convolution of ultra-logconcave sequences,” Electron. J. Combin., 16, Note 5 (2009).
H. Hadwiger, Vorlesungen über Inhalt, Oberflache, und Isoperimetrie, Springer Verlag, Berlin (1957).
H. Hadwiger, “Das Wills’sche Funktional,” Monatsh. Math., 79, 213–221 (1975).
H. Hadwiger, “Gitterpunktanzahl im Simplex und Willssche Vermutung,” Math. Ann., 239, 271–288 (1979).
K. Itô and M. Nisio, “On the oscillation functions of Gaussian processes,” Math. Scand., 22, 209–223, 1968 (1969).
D. A. Klain, “A short proof of Hadwiger’s characterization theorem,” Mathematika, 42, 329–339 (1995).
H. Le, “On bounded Gaussian processes,” Statist. Probab. Lett., 78, 669–674 (2008).
M. Ledoux, The Concentration of Measure Phenomenon, Amer. Math. Soc., Providence (2001).
M. Ledoux and M. Talagrand, Probability in Banach Spaces, Springer-Verlag, New York (1991).
M. Lifshits, Gaussian Random Functions, Kluwer, Boston (1995).
T. M. Liggett, “Ultra logconcave sequences and negative dependence,” J. Combin. Theory Ser. A, 79, 315–325 (1997).
P. McMullen, “Non-linear angle-sum relations for polyhedral cones and polytopes,” Math. Proc. Cambridge Philos. Soc., 78, 247–261 (1975).
P. McMullen, “Inequalities between intrinsic volumes,” Monatsh. Math., 111, 47–53 (1991).
R. Pemantle, “Towards a theory of negative dependence. Probabilistic techniques in equilibrium and nonequilibrium statistical physics,” J. Math. Phys., 41, 1371–1390 (2000).
R. Schneider, Convex Bodies: the Brunn–Minkowski Theory, 2nd ed., Cambridge Univ. Press, New York (2014).
G. C. Shephard, “Inequalities between mixed volumes of convex sets,” Mathematika, 7, 125–138 (1960).
V. N. Sudakov, “Gaussian random processes and the measures of solid angles in Hilbert space,” Dokl. Akad. Nauk SSSR, 197, 43–45 (1971).
V. N. Sudakov, “Geometric problems of the theory of infinite-dimensional probability distributions,” Trudy Mat. Inst. Steklov, 141 (1976).
V. N. Sudakov, “Geometric problems in the theory of infinite-dimensional probability distributions,” Cover to cover translation of Trudy Mat. Inst. Steklov, 141 (1976), Proc. Steklov Inst. Math., No. 2, 1–178 (1979).
B. S. Tsirel’son, “A geometric approach to maximum likelihood estimation for infinite-dimensional Gaussian location. I,” Theory Prob. Appl., 27, 411–418 (1982).
B. S. Tsirel’son, “A geometric approach to maximum likelihood estimation for infinite-dimensional Gaussian location. II,” Theory Prob. Appl., 30, 820–828 (1985).
B. S. Tsirel’son, “A geometric approach to maximum likelihood estimation for infinite-dimensional location. III,” Theory Prob. Appl., 31, 470–483 (1986).
R. A. Vitale, “The Wills functional and Gaussian processes,” Ann. Probab., 24, 2172–2178 (1996).
R. A. Vitale, “A log-concavity proof for a Gaussian exponential bound,” in: T.P. Hill and C. Houdré (eds.) Contemporary Math.: Advances in Stochastic Inequalities, 234, Amer. Math. Soc. (1999), pp. 209–212.
R. A. Vitale, “Intrinsic volumes and Gaussian processes,” Adv. Appl. Prob., 33, 354–364 (2001).
R. A. Vitale, “A question of geometry and probability,” in: A Festschrift for Herman Rubin, IMS Lecture Notes Monogr. Ser., 45 (2004), pp. 337–341.
R. A. Vitale, “On the Gaussian representation of intrinsic volumes,” Statist. Probab. Lett. 78, 1246–1249 (2008).
J. M. Wills, “Zur Gitterpunktanzahl konvexer Mengen,” Elemente der Math., 28, 57–63 (1973).
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Published in Zapiski Nauchnykh Seminarov POMI, Vol. 457, 2017, pp. 101–113.
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Vitale, R.A. On an Exponential Functional for Gaussian Processes and Its Geometric Foundations. J Math Sci 238, 406–414 (2019). https://doi.org/10.1007/s10958-019-04247-4
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DOI: https://doi.org/10.1007/s10958-019-04247-4