Convexity: An Analytic Viewpoint.

1. Convex functions and sets — p. 1
   1.1 Convex subsets of real vector spaces
   1.2 Convex combinations and convex hulls
   1.3 Convex and strictly convex functions
   1.4 Epigraphs and geometric interpretation
   1.5 One-dimensional convexity and secant slopes
   1.6 Continuity and differentiability of convex functions
   1.7 Supporting lines and tangent inequalities
   1.8 Jensen’s inequality
   1.9 Hölder and Minkowski inequalities from convexity
   1.10 Gauges of convex sets
   1.11 Legendre transforms
   1.12 Hahn–Banach viewed through supporting tangents
   Simon explicitly identifies differentiability, Jensen,
   gauges, Legendre transforms, and the Hahn–Banach viewpoint as
   foundational themes  

2. Orlicz spaces
   2.1 Young functions
   2.2 Convex growth functions beyond power laws

• 2.3 Modular functionals
  2.4 Luxemburg-type norms
  2.5 Orlicz classes and Orlicz spaces
  2.6 Hölder-type inequalities in Orlicz spaces
  2.7 Complementary Young functions
  2.8 Duality phenomena
  2.9 Comparison with Lp spaces
  2.10 Convexity as the mechanism generating function spaces

1. Gauges and locally convex spaces — p. 51
   3.1 Minkowski gauge of a convex absorbing set
   3.2 Balanced and absolutely convex sets
   3.3 Seminorms generated by gauges
   3.4 Families of seminorms
   3.5 Locally convex vector spaces
   3.6 Neighborhood bases at the origin
   3.7 Continuous linear functionals
   3.8 Metrizability and completeness issues
   3.9 Banach spaces as a special locally convex case
   3.10 Geometry-topology correspondence through convex neighborhoods

2. Separation theorems — p. 66
   4.1 Hyperplane separation
   4.2 Point-versus-convex-set separation
   4.3 Separation of disjoint convex sets
   4.4 Strong versus weak separation
   4.5 Closedness assumptions
   4.6 Continuous linear functionals as separators
   4.7 Hahn–Banach separation
   4.8 Geometric meaning of dual functionals
   4.9 Separation as the bridge from convex geometry to duality

3. Duality: dual topologies, bipolar sets,
   and Legendre transforms — p. 70
   5.1 Algebraic and continuous duals
   5.2 Weak topology
   5.3 Weak-star topology
   5.4 Polar sets
   5.5 Bipolar constructions
   5.6 Bipolar theorem
   5.7 Dual descriptions of convex sets
   5.8 Convex conjugation
   5.9 Legendre–Fenchel transform
   5.10 Double conjugation and recovery
   5.11 Supporting functionals and subgradients
   5.12 Duality as reconstruction of convex structure

4. Monotone and convex matrix functions  
   6.1 Hermitian matrices and matrix order
   6.2 Positive matrices
   6.3 Scalar versus matrix monotonicity
   6.4 Matrix monotone functions
   6.5 Matrix convex functions
   6.6 Spectral functional calculus
   6.7 Operator inequalities
   6.8 Divided differences
   6.9 Positivity criteria
   6.10 Why ordinary scalar convexity is insufficient in matrix order
   6.11 Preparation for Loewner theory

5. Loewner’s theorem: a first proof 
   7.1 Operator-monotone functions
   7.2 Loewner matrices
   7.3 Positivity of divided-difference matrices
   7.4 Necessary conditions for operator monotonicity
   7.5 Analytic continuation
   7.6 Pick/Nevanlinna-type analytic structure
   7.7 Integral representations
   7.8 First proof of Loewner’s characterization
   7.9 Matrix order as an analytic constraint

6. Extreme points and the Krein–Milman theorem  
   8.1 Extreme points
   8.2 Faces of convex sets
   8.3 Convex hulls of extreme points
   8.4 Compact convex sets
   8.5 Finite-dimensional intuition
   8.6 Infinite-dimensional complications
   8.7 Krein–Milman theorem
   8.8 Existence of extreme points
   8.9 Examples in function and measure spaces
   8.10 Extreme points as minimal generators of convex structure

7. The Strong Krein–Milman theorem  
   9.1 Strengthening ordinary Krein–Milman
   9.2 Closed convex hull reconstruction
   9.3 Exposed versus extreme points
   9.4 Faces and supporting hyperplanes
   9.5 Boundary structure of compact convex sets
   9.6 Strong generation statements
   9.7 Stability under convex reconstruction
   9.8 Transition from geometry to representing measures

8. Choquet theory: existence 
   10.1 Barycenters
   10.2 Probability measures on compact convex sets
   10.3 Representing a point by a measure
   10.4 Measures supported on extreme points
   10.5 Choquet representation
   10.6 Existence of representing measures
   10.7 Approximation by finite convex combinations
   10.8 Boundary concentration
   10.9 Krein–Milman versus Choquet: points versus measures

9. Choquet theory: uniqueness — p. 171
   11.1 Nonuniqueness of convex decompositions
   11.2 Simplexes
   11.3 Choquet simplexes
   11.4 Uniqueness of representing measures
   11.5 Maximal measures
   11.6 Boundary measures
   11.7 Affine functions as probes of representation
   11.8 When extreme-point data determine the whole convex object
   11.9 Structural meaning of unique barycentric decomposition

10. Complex interpolation 
    12.1 Compatible pairs of Banach spaces
    12.2 Analytic families of vectors and operators
    12.3 Strip method
    12.4 Boundary norm control
    12.5 Three-lines type convexity
    12.6 Interpolation norms
    12.7 Riesz–Thorin-type consequences
    12.8 Interpolation of operators
    12.9 Log-convexity of norms
    12.10 Convexity hidden inside analytic interpolation

11. Brunn–Minkowski inequalities and log-concave functions  
    13.1 Minkowski addition of sets
    13.2 Volume under addition
    13.3 Brunn–Minkowski inequality
    13.4 Concavity of volume powers
    13.5 Log-concave functions
    13.6 Functional forms of Brunn–Minkowski
    13.7 Prékopa-type phenomena
    13.8 Level sets of log-concave functions
    13.9 Convex geometry versus integral inequalities
    13.10 Applications to probability and analysis

12. Rearrangement inequalities I:
    Brascamp–Lieb–Luttinger  
    14.1 Symmetric decreasing rearrangement
    14.2 Equimeasurable functions
    14.3 Rearrangement of sets
    14.4 Preservation of distribution functions
    14.5 Integral comparison principles
    14.6 Brascamp–Lieb–Luttinger inequality
    14.7 Multiple-integral rearrangements
    14.8 Symmetrization
    14.9 Extremizers and geometric concentration
    14.10 Rearrangement as order improvement without changing mass

13. Rearrangement inequalities II:
    Majorization 
    15.1 Majorization of vectors
    15.2 Doubly stochastic transformations
    15.3 Permutation averaging
    15.4 Hardy–Littlewood–Pólya theory
    15.5 Schur-convex functions
    15.6 Karamata-type inequalities
    15.7 Matrix majorization
    15.8 Eigenvalue inequalities
    15.9 Function majorization
    15.10 Rearrangement, convex order, and entropy
    15.11 Majorization as a partial order generated by mixing

14. The relative entropy 
    16.1 Entropy and convexity
    16.2 Classical relative entropy
    16.3 Matrix/quantum relative entropy
    16.4 Convexity properties
    16.5 Positivity
    16.6 Klein-type inequalities
    16.7 Variational principles
    16.8 Monotonicity under admissible transformations
    16.9 Entropy as a convex divergence
    16.10 Connections with matrix convexity and majorization

15. Notes  
    17.1 Historical origins of convexity
    17.2 Gibbs and thermodynamic convexity
    17.3 Development of functional-analytic convexity
    17.4 Hahn–Banach and separation
    17.5 Legendre duality
    17.6 Orlicz-space history
    17.7 Loewner and matrix-monotone functions
    17.8 Krein–Milman and Choquet theory
    17.9 Brunn–Minkowski and rearrangement traditions
    17.10 Entropy and mathematical physics 

References  
Author index  
Subject index 

A useful higher-level organization is therefore:

Part I — Foundations
Chapter 1

Part II — Convexity, topology, and duality in
infinite-dimensional spaces
Chapters 2–5

Part III — Matrix convexity and Loewner theory
Chapters 6–7

Part IV — Extreme structure and representation
Chapters 8–11

Part V — Convexity as an engine for analytic inequalities
Chapters 12–16

Part VI — Historical and conceptual synthesis
Chapter 17

One important boundary: the algebraic-geometric notion of “convex
variety” in the Wikipedia link is not the same notion as Simon’s convex
sets/functions. In algebraic geometry it is formulated through unobstructed
rational curves, commonly via vanishing of H¹(C, f*T_X); projective spaces,
homogeneous spaces such as G/P, products, and certain projective bundles
are examples. Its main role is deformation theory and the geometry of
Kontsevich moduli spaces. (Wikipedia)

Appendix A — Convexity beyond convex analysis
A.1 Euclidean/functional convexity versus algebraic-geometric convexity
A.2 Rational curves and tangent-bundle pullback
A.3 Vanishing of obstruction spaces
A.4 Projective-space examples
A.5 Homogeneous spaces and Grassmannians
A.6 Products and projective bundles
A.7 Deformation theory
A.8 Kontsevich moduli spaces
A.9 Convexity as “absence of obstruction” rather than
“closure under line segments”
A.10 Structural comparison of the two notions.