These lecture s wer e prepare d fo r th e advance d undergraduat
in Geometric Combinatorics a t th e Par k Cit y Mathematic s
in Jul y 2004 . Man y thank s t o th e organizer s o f th e unde
ate program , Bil l Barke r an d Roge r Howe , fo r invitin g m e
this course . I als o wis h t o than k Ezr a Miller , Vi c Reine r an
Sturmfels, wh o coordinate d th e graduat e researc h progra m a
for thei r support . Edwi n O'She a conducte d al l th e tutorial
course and wrote several of the exercises seen in these lectures
was a hug e hel p i n th e preparatio n o f thes e lecture s fro m be
to end .
The mai n goa l of these lecture s wa s to develo p th e theor
vex polytope s fro m a geometri c viewpoin t t o lea d u p t o rece
opments centere d aroun d secondar y an d stat e polytope s aris
point configurations . Th e geometri c viewpoint naturall y relie
ear optimizatio n ove r polytopes . Chapter s 2 and 3 develop t
of polytop e theory . I n Chapter s 4 an d 5 we see th e tool s o f
and Gal e diagram s fo r visualizin g polytope s an d understandi
facial structure . Gal e diagram s hav e bee n use d t o uneart h
bizarre phenomen a i n polytopes , suc h a s th e existenc e o f p
whose vertice s canno t hav e rationa l coordinate s an d other
facets canno t b e prescribed . Thes e example s ar e describe d i
ter 6 . I n Chapter s 7- 9 w e construc t th e secondar y polyto
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