Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Synthetic MTW conditions and their equivalence under mild regularity
  assumption on the cost function   Loeper's condition in \cite{Loe09} and the quantitatively quasi-convex
condition (QQconv) from \cite{GK15} are synthetic expressions of the analytic
MTW condition from \cite{TW} since they only require $C^2$ differentiability of
the cost function $c$. When the cost function $c$ is $C^4$, it is known that
the two synthetic MTW conditions are equivalent to the analytic MTW condition.
However, when the cost function has regularity weaker than $C^4$, it is not
known that if the two synthetic MTW conditions are equivalent. In this paper,
we show the equivalence of the synthetic MTW conditions when the cost function
has regularity slightly weaker than $C^3$.

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The categories of this paper are: astro-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: hep-th
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The categories of this paper are: hep-lat
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The categories of this paper are: dg-ga
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The categories of this paper are: econ
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The categories of this paper are: stat
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The categories of this paper are: cmp-lg
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The categories of this paper are: q-fin
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The categories of this paper are: physics
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The categories of this paper are: mtrl-th
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The categories of this paper are: cond-mat
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The categories of this paper are: nucl-ex
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The categories of this paper are: supr-con
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The categories of this paper are: alg-geom
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The categories of this paper are: chao-dyn