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\section{1}
the optimal solution for some objective function subject to a set of
inequalities 
\section{2}
the extremal point of a polygon defined by the intersection of $n$
half-spaces in $d$-dimensional space
\section{3}
mention ellipsoid, interior point
\section{4}
\section{5}
mention d constraints define a point so a basis has size d

mention we assume we start with a basis
\section{6}
\section{7}
mention very similar to seidel if we take r = n-1
\section{8}
motivate we only augment G d times
\section{9}
\section{10}
\section{11}
\section{12}
mention successful iteration is one in which state changes

analyze with parameter r and optimize running time for r later
\section{13}
mention violated constraint -> violated basis constraint
\section{14}
\section{15}
mention this removes linear dependence on n in part exp in d
\section{16}
mention bottleneck in running time of last alg was recursive call
because that's exponential in input size, so adding a poly number
of loops while reducing input to recursion is good for us

mention every time a constraint is violated, a basis constraint is
violated, so the probability we include the basis constraints in the
random sample grows really fast
\section{17}
mention |H| includes multiplicity

for all constraints in $V$, \hfill~\\[-1ex]
double their multiplicity in $H$\hfill~\\[-1ex]
\section{18}
mention a successful iteration is one in which the state changes
\section{19}
mention in every successful iteration, some basis constraint is
violated 

mention the multiplicity of some constraint in the basis is doubled
\section{20}
\section{21}
mention if we solve lp for $n = O(d^2)$ in pol time, we get pol time alg
for lp in general
\section{22}
mention this removes poly dependence on n in part exp in d
\section{23}
mention when we project constraints, we disregard all information
about the basis, but it might help us.  that is, the old solution
is probably close to the new solution; they probably share basis
constraints
\section{24}

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