Preface These lecture s wer e prepared fo r th e advance d undergraduat e cours e in Geometric Combinatorics a t th e Par k Cit y Mathematic s Institut e in Jul y 2004 . Man y thank s t o th e organizer s o f th e undergradu - ate program , Bil l Barke r an d Roge r Howe , fo r invitin g m e t o teac h this course . I als o wish t o than k Ezr a Miller , Vi c Reine r an d Bern d Sturmfels, wh o coordinated th e graduate researc h program a t PCMI , for thei r support . Edwi n O'She a conducte d al l th e tutorial s a t th e course and wrote several of the exercises seen in these lectures. Edwi n was a hug e hel p i n th e preparatio n o f thes e lecture s fro m beginnin g to end . The mai n goa l of these lectures was to develop the theory o f con- vex polytopes fro m a geometric viewpoin t t o lea d u p t o recen t devel - opments centere d aroun d secondar y an d stat e polytope s arisin g fro m point configurations . Th e geometric viewpoint naturall y relie s on lin- ear optimizatio n ove r polytopes. Chapter s 2 and 3 develop the basic s of polytop e theory . I n Chapter s 4 and 5 we see the tool s o f Schlege l and Gal e diagrams fo r visualizin g polytopes an d understandin g thei r facial structure . Gal e diagram s hav e bee n use d t o uneart h severa l bizarre phenomen a i n polytopes , suc h a s th e existenc e o f polytope s whose vertice s canno t hav e rationa l coordinate s an d other s whos e facets canno t b e prescribed . Thes e example s ar e describe d i n Chap - ter 6 . I n Chapter s 7- 9 w e construc t th e secondar y polytop e o f a vii
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