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The stability is reflected in the rate of change of cβ for β bounded away from 1, which one can estimate by standard volume estimates on the sphere. Thus, cβ+δ < cβ−δ (1 + Cδ). This is what we consider a stability result. We remark that it is not difficult to check that for, say, β > 1/2 and bounded away from 1, we have c0 cβ < cβ < c0 Cβ and thus c C Dn ⊂ K(β) ⊂ Dn . ) The same is true for β < 1/2 and bounded away from 0. The reason that stability results can be important is that sometimes one cannot check exactly if a proportion 1/2 of the inequalities is fulfilled, but can do the following weaker thing: to have a set so that each point in the set satisfies at least 1/2 − δ of the inequalities, and each point outside the set has at least 1/2 − δ inequalities which it violates.

Let us denote by φ˜ the image of φ ˜ is Wi /Wi+1 = C ∞ (X, Valsm i (T X)). Thus φ = 0. We will show that there exists ψ ∈ Wn−i,c such that φ · ψ = 0. 3(2). Thus it is enough to show that for any φ˜ ∈ C ∞ (X, Valsm i (T X)) there exists ψ˜ ∈ Cc∞ (X, Valsm n−i (T X)) such that φ˜ · ψ˜ = 0 X where the product φ˜ · ψ˜ is understood pointwise in the tangent space of each ∞ point, φ˜ · ψ˜ ∈ Cc∞ (X, Valsm n (T X)) = Cc (X, |ωX |), and the integration is understood in the sense of the usual integration of densities.

We wish to find vectors ε(1), . . , ε(N ) such that 1 N N | ε(j), x | |x|. (4) j=1 Notice that, obviously, this cannot be achieved by ≤ n vectors since this (1+δ)n does would give an embedding of n2 into n1 . However, as we know that 1 have isomorphic euclidean sections of dimension n (see [K]), it is conceivable that such an embedding can be constructed with a matrix of random signs. Geometric Applications of Chernoff-Type Estimates 53 This problem has a history. It was first shown by Schechtman in [S2] that the above is possible with a random selection of N = Cn vectors, where C is a universal constant, and then repeated in [BLM] in a more general context including Kahane-type generalization.