  Given a convex function $f$ on $\mathbb{R}^n$ with an integer minimizer, we
show how to find an exact minimizer of $f$ using $O(n^2 \log n)$ calls to a
separation oracle and $O(n^4 \log n)$ time. The previous best polynomial time
algorithm for this problem given in [Jiang, SODA 2021, JACM 2022] achieves
$O(n^2\log\log n/\log n)$ oracle complexity. However, the overall runtime of
Jiang's algorithm is at least $\widetilde{\Omega}(n^8)$, due to expensive
sub-routines such as the Lenstra-Lenstra-Lov\'asz (LLL) algorithm [Lenstra,
Lenstra, Lov\'asz, Math. Ann. 1982] and random walk based cutting plane method
[Bertsimas, Vempala, JACM 2004]. Our significant speedup is obtained by a
nontrivial combination of a faster version of the LLL algorithm due to
[Neumaier, Stehl\'e, ISSAC 2016] that gives similar guarantees, the volumetric
center cutting plane method (CPM) by [Vaidya, FOCS 1989] and its fast
implementation given in [Jiang, Lee, Song, Wong, STOC 2020].
  For the special case of submodular function minimization (SFM), our result
implies a strongly polynomial time algorithm for this problem using $O(n^3 \log
n)$ calls to an evaluation oracle and $O(n^4 \log n)$ additional arithmetic
operations. Both the oracle complexity and the number of arithmetic operations
of our more general algorithm are better than the previous best-known runtime
algorithms for this specific problem given in [Lee, Sidford, Wong, FOCS 2015]
and [Dadush, V\'egh, Zambelli, SODA 2018, MOR 2021].
