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Let $\Omega$ be a bounded open set in $R^{n}$ and $K$ compact in $R^{m}$. Consider $L^{p}(\Omega,R^{m})$ defined as a vector valued $f=(f_{1},\cdots,f_{m})$ where $f_{i}\in L^{p}$ . Let $v_{j}: \Omega \to R^{m}$ be such that $$ v_{j} \to^{\star} v \ \text{in} \ L^{\infty }(\Omega) \ \text{and} \ \ v_{j}(x)\in K \ \ a.e $$ The convergence above is weakly star. I need to prove that $v(x)\in \overline{conv(K)}$.

I wish you can help me.

I also have problem with proving the converse, if $v(x)\in \overline{conv(K)}$ , show that there exists a sequence $\lbrace v_{j} \rbrace$ such that $$ v_{j}\to ^{\star} v \ \ \text{in} \ L^{\infty}(\Omega) \ \ \text{and} \ v_{j}(x)\in K$$

Thanks you all for your help!!

user105554
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1 Answers1

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For the first question, recall that a closed convex set is an intersection of closed half-spaces, and use the weak convergence.

For the converse, consider the weak* $L^\infty$ closure of the set of all functions valued in $K$. It is included in the set of all $L^\infty$ functions valued in $\mathrm{\overline{ co}}(K)$ by the first part, and you only need to prove it contains all $\mathrm{{ co}}(K)$-valued simple functions; or also, by a localizing argument, all $\mathrm{{ co}}(K)$-valued constant functions. But it is easy to realize a convex combination of points of $K$ as a weak* limit of a sequence of $K$-valued simple functions. To this end, the observation is that any constant function $0\le c \le 1$, is indeed a weak* limit of a sequence of characteristic functions of subsets of $\Omega$.

Pietro Majer
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