By Pierre de la Harpe

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Sequence I IX-Xol I ~ 6. Let {~n }. Choose x n = 6y n. Choose Yn c S(X) Then we o b t a i n so the contradiction i ~ f(xo+Xn)-f(xo) K ~(Xn) : 6¢(y n) ~ 6(I[¢nll l/n). Example. Let X For the (continuous) in 6a), we see that are c o n t a i n e d §9. Normal is, t > 0. convex if that K if at H closed The x° described (linearly S(xo,K ) (that is, of Yo = llR(x)-Yo] [, of f support of x to c Y. introduced at any point independent) is the cone K; to that which O K at in in K X K. this supporting intersection E Ks V points.

Kuhn-Tucke r Theory As a second illustration of the use of lld), we consider a special class of convex programs in the finite dimensional ("ordinary convex programs") case, have been of great practical and for which an elegant theory is available. intuitively described as "minimizing which, interest, The programs may be a convex function subject to convex constraints". a) Lemma. ,K n be closed convex bodies whose interiors have a point in common. N(Xo,K) = Let in a ics x ° s K ~ ( ~ K i. X Then ~N(Xo,Ki).

K. hyperplanes containing from x ° ~ M, subset is the it is the of all is a m e a s u r e at each a convex denoted When emanating vertex of its b o u n d a r y x° contain xo, cone at each is, a cone w i t h x ~ K~xo+t(X-Xo) at of the union since the h a l f - s p a c e s K X, of rays is a convex cone a point Xo, then be a ics, S(xo,K ) at xo, M by at cone through K at X to a is is a union is a convex as the closure the d i m e n s i o n belongs xo) Let cones support and p a s s i n g boundary at generated convex The x° a subspace a) Definition.