Preface ......................................................... v
Notation ...................................................... vii
0 Prerequisites ................................................ 1
1 The affine structure of  n .................................. 11
1.1 Planes ................................................. 11
1.2 Affine spans ........................................... 37
1.3 Affine transformations ................................. 57
2 Convex sets ................................................. 71
2.1 Algebraic properties of convex sets .................... 71
2.2 Relative interior of convex sets ....................... 78
2.3 Closure and relative boundary of convex sets ........... 93
3 Convex hulls ............................................... 117
3.1 Algebraic properties of convex hulls .................. 117
3.2 Topological properties of convex hulls ................ 128
4 Convex cones and conic hulls ............................... 145
4.1 Convex cones .......................................... 145
4.2 Conic hulls ........................................... 161
5 Recession and normal directions ............................ 183
5.1 Recession cones ....................................... 183
5.2 Linearity spaces ...................................... 197
5.3 Normal and barrier cones .............................. 206
6 Support and separation properties .......................... 227
6.1 Bounds, supports, and asymptotes ...................... 227
6.2 Containing and supporting halfspaces .................. 235
6.3 Separation by hyperplanes and slabs ................... 241
7 The extreme structure of convex sets ....................... 255
7.1 Extreme faces ......................................... 255
7.2 Extreme representations ............................... 267
8 The exposed structure of convex sets ....................... 287
8.1 Exposed faces ......................................... 287
8.2 Exposed representations ............................... 299
9 Polyhedra .................................................. 319
9.1 Intersections of halfspaces ........................... 319
9.2 Properties of polyhedra ............................... 328
Solutions to Exercises ..................................... 345
Bibliography .................................................. 387
Author Index .................................................. 399
Subject Index ................................................. 403
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