Notions of Convexity Book

	Preface
Ch. I	Convex functions of one variable
	Definitions and basic facts	1
	Some basic inequalities	9
	Conjugate convex functions (Legendre transforms)	16
	The [Gamma] function and a difference equation	20
	Integral representation of convex functions	23
	Semi-convex and quasi-convex functions	26
	Convexity of the minimum of a one parameter family of functions	28
Ch. II	Convexity in a finite-dimensional vector space
	Definitions and basic facts	36
	The Legendre transformation	66
	Geometric inequalities	75
	Smoothness of convex sets	94
	Projective convexity	98
	Convexity in Fourier analysis	111
Ch. III	Subharmonic functions
	Harmonic functions	116
	Basic facts on subharmonic functions	141
	Harmonic migrants and the Riesz representation formula	171
	Exceptional sets	203
Ch. IV	Plurisubharmonic functions
	Basic facts	225
	Existence theorems in L[superscript 2] spaces with weights	248
	Lelong numbers of Lelong numbers of plurisubharmonic functions	265
	Closed positive currents	271
	Exceptional sets	285
	Other convexity conditions	290
	Analytic functionals	300
Ch. V	Convexity with respect to a linear group
	Smooth functions in the whole space	315
	General G subharmonic functions	324
Ch. VI	Convexity with respect to differential operators
	P-convexity	328
	An existence theorem in pseudoconvex domains	332
	Analytic differential equations	344
Ch. VII	Convexity and condition [Psi]
	Local analytic solvability for [actual symbol not reproducible]	353
	Generalities on projections and distance functions, and a theorem of Trepreau	372
	The symplectic point of view	375
	The microlocal transformation theory	382
	App. A. Polynomials and multilinear forms	391
	App. B. Commutator identities	396
	Notes	403
	References	407
	Index of notation	411
	Index	413