# Convex Analysis and Measurable Multifunctions by Charles Castaing

By Charles Castaing

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Additional info for Convex Analysis and Measurable Multifunctions

Sample text

1) Let f : 1 f (x)E R and x,~ E'. Then the following 0 - 25 a) x, E of (x ) o = < x', xo> b) f*(x') +. f(X O) c) f*(x') + f(x ) S o Consequently of (x ) x >. 0 is closed and convex. o 2) Let = o*(Y'\C)}. >o f(x ) ~ f*(x') o x > ~ f*(x') + f(x ). o 0 a ~ c. ) x'E E', f*(x') ~ - f(x ) o 0 b .. c. Finally of (x o ) is a(E', E) = (x' E E'I- + f*(x')S -f(x 0 )} 0 closed and convex.

E(A,B) A C C B + eU e > e(A, B) B +- eU} = e(A , B) . s. 50 = s up sup x E A x' E UO s up x EA x 'EUo sup =. = "up x' Ello inf y E B [ - 5*(x'iB)] [6*(x'IA) - 6*(x'IB)]. ding Definition. Let homogeneous of E', ~ q:>c b\E) in a vector space. continuous sets K are counded. and strongly contin uous. With the topology of unifor m convergence on eq ui(;outinuous sets, }e becomes a Haus dor ff lo call-y convex vector spac e. The orem to 7l{ II-19.. le t e . The mapping from ~ 6* (. IA)

In A c U. then AC is VO : k}. AO ~ co (x;, ••• , x From r(t o ) c colx1 , •• • , fo llows Xn } + J i 0 6*(xjl r(t» Then 2V cAe U 6*(x' l r(t)),; sup 6*( x'jx . J V be a neighbourhood of Let 1 ,; 1 for t t ret) cAe U for every 21 - Theorem II- ::> l. Let T r convex sre ce and f:: +; J such tilat o E V, j t V) < 6*(x'IA ) ,; 1- 1 •••• , k. E V. We suppose t l. t~T ret) to ta l l y bou nded. )) are t . c. at ~. We say that whi ch mee ts ret) n u ~ ~ r (t) r(t) o r is l. c. a t mee ts the open set 5*(x'lr(t)) o \ i f for any open s e t = -~ V of t 0 U such t hat t E V.