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By Richard B. Holmes (auth.)

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For which the limit in on which the value of (i) de[26, p. 97]. With the same hypotheses, Sf(Xo) consists Exercise d) Remarks. i) is @ ~ ¢(x) ~ z Vf(Xo) exists if of a single element, namely ~. 18. An alternative proof of the theorem in b) will be given later, in §18, to illustrate a formula involving conjugate functions. 2) Since we will frequently convex function f be continuous encounter the hypothesis at a point, recall that this is not a very stringent that a it is worthwhile requirement. Namely, is bounded above on some non-empty open set, in particular, upper semi-continuous (usc) at any point, continuous int throughout (dom (f)) f ~ Conv continuous (X) throughout where rel-int X (dom (f)).

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. of closed X, convex intersection S(xo~K), rays of all x° can also be emanating is a support of the smoothness in turn depends of the supporting K x o. 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.

The properties: (f+c)* = f*-c, (cf) * = cf*(c), f*(y) = g*(cy) f*(Y) = g~(y+F) c s R I. c > O; if f(x) if f(x) = g(x/c), = g(x)- c > 0; If generally, X of for some M K C . X, This then function is called K. be a nls and if X. : x ~ K} ~ sup < K , y ~ function Let f = 6K on f(x) is a subspace = jJxll. of X, Then and f* = 6U(X, ). f(x) = dist (x,M), 44 then f* = 6U(Mi ). 3) Then Let f*(Y) 4) X be a nls and for = I]yl ]q/q, Let X = R1 where and f(x) = e x n n × Let X = Rn matrix.

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