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1 Also det ~x,y)=(x-y)2+ ½ > 0 V (x,y) E R 2. 1 Nowever J(x,y)is a P-matrix if x 2 y 2 > ½. Consequently F satisfies the conditions of theorem 2 and hence F is one-one throughout R 2 . Mas-Colell's univalent result : Theorem 3 : We will now prove the following theorem of Mas-Colell. Let F:~ ÷ R n be a continuously differentiable function where ~ is a compact convex polyhedron of full dimension. For every nonempty subspace let ~L: Rn + L denote the perpendicular projection map. If for every x s L~R n ~ and subspace L C R n spanned by a face of K (that is, the translation to the origin of the minimal affine space containing K)which includes x, the map H L DF(x) : L ÷ L has a positive determinant (that is the linear map H L- DF(x) preserves orientation where DF(x) stands for the derivative map of F at x) then F is one-one on ~ and consequently a homeomorphism.

We will give two different proofs one due to Garcia and Zangwill and the other due to Mas-Colell. Proof of Carcia and Zangwill uses the norm-coerciveness theorem whereas Mas-Colell uses results from degree theory. There is a subtle difference between these results and Cale-Nikaido's theorem. fundamental The difference lies in the J fact that the proofs of Garcia-Zangwill whereas Gale-Nikaido's a C (I) fanction. It is not clear whether Garcia-Zangwill result holds good if we assume F to be a differentiable This seems to be an interesting open problem in this area.

We will now prove that case Since 61 as well as 62 can be chosen as small as we please, we will choose them so small so that both x and y are more than a distance of 6 from the boundary. G(x) = F(x), G(y) = F(y) and F(x) = F(y). and it is norm coercive. This will imply G(x) = G(y) since Clearly G is continuously differentiable If x s ~ is at least a distance of 6 from ~ , Jacobian of G(x) = Jacobian of F and hence det of the Jacobian of G(x) is positive. is within 6 of If x 3~, Jacobian G(x) is a P-matrix and hence its determinant is positive.

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An Integrated Approach to Image Watermarking and JPEG-2000 Compression by Su Wang Kuo


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