# Upper bounds for $NP$ based on $NEXP = EXP$

It's open whether $EXP = NEXP \to P = NP$ (the other direction can be shown by padding). My question: has there been any progress along these lines at all? For example, can we show that $EXP = NEXP \to NP \subset SUBEXP$ (where $SUBEXP = \cap_{\epsilon > 0} 2^{n^\epsilon}$)?

• Perhaps you will get more responses in cstheory.se. – Yuval Filmus Dec 17 '15 at 18:09
• this reminds me of exponential time hypothesis which is probably "close/ related"... – vzn Dec 17 '15 at 20:07

If someone has demonstrated that $$EXP=NEXP\implies NP\subseteq SUBEXP$$, then that guy has unconditionally proved a long-sought separation $$P^{NP}\subsetneq NEXP$$.
Proof: If $$EXP\neq NEXP$$, then obviously, $$P^{NP}\neq NEXP$$.
So assuming $$EXP=NEXP$$, the above statement gives us $$NP\subseteq SUBEXP$$. We have:
$$P^{SUBEXP}=SUBEXP$$
So, $$P^{NP}\subseteq SUBEXP\subsetneq EXP$$.$$\blacksquare$$