Grothendieck's inequality ( http://en.wikipedia.org/wiki/Grothendieck_inequality
) asserts that a certain discrete optimisation problem (optimising a bilinear form over inputs that are +1 or -1) is equivalent up to constants to a continuous optimisation problem (optimising the same bilinear form over unit vectors in a Hilbert space). This is important for theoretical computer science, because the former problem essentially contains NP-hard problems such as MAX-CUT, whereas the latter can be rephrased as a semidefinite program (by writing the problem in terms of the Gram matrix of the unit vectors) and can be solved in polynomial time by algorithms such as the ellipsoid method. So there are certain NP-hard problems which one can solve "up to constants" in polynomial time: in some sense, the "ratio" between NP and P is bounded! Furthermore, in a certain technical sense, one cannot achieve a better approximation to such problems by any
polynomial-time algorithm than through the Grothendieck inequality, at least if one assumes the Unique Games Conjecture (a result of Raghavendra and Steurer).
Grothendieck's inequality also has a very cute connection to Bell's inequality in quantum mechanics, as observed by Tsirelson; roughly speaking, the ability of quantum mechanics to violate Bell's inequality is logically equivalent to the constant in Grothendieck's inequality being greater than one (basically because the discrete optimisation problem describes the envelope of all possible measurement outcomes of a classical hidden-variable system, and the vector-valued optimisation problem describes the envelope of all possible measurement outcomes of a quantum system). Or to put it another way, Grothendieck's inequality asserts (in some sense) that Bell's inequality can only be violated "up to a constant", at least when there are only two measurements made. (For three or more measurements the violation can be much more dramatic, a result of Perez-Garcia, Wolf, Palazuelos, Villenueva, and Junge.)
[All this I learned today from a very nice lecture by Pisier on these topics, largely based on his survey article linked to here.] #spnetwork #recommend