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Can anyone help me to access the paper:

M.S Brodskii and D.P Milman, "On the center of a convex set", Dokl. Akad. Nauk SSSR 59 (1948) 837–840 in Russian?

or to prove the theorem:

If $K$ is a weakly compact, bounded convex subset of a Banach space with normal structure, then there exists $x_0\in K$ such that $T(x_0)=x_0$ for all surjective isometry $T:K→K$. That is, there is a common fixed point for the family of surjective isometries on $K$.

Thank you.

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  • $\begingroup$ Excuse my ignorance (Google didn't help me), but would you define normal structure for Banach spaces? (BTW, would the group of isometries be compact?). $\endgroup$ Sep 10, 2014 at 6:57
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    $\begingroup$ @Włodzimierz Holsztyński Google helped me a bit: see projecteuclid.org/euclid.pjm/1102985731 or the free pdf projecteuclid.org/download/pdf_1/euclid.pjm/1102985731 . $\endgroup$
    – TaQ
    Sep 10, 2014 at 10:45
  • $\begingroup$ Thank you. This link helped me to enhance my knowledge of normal structure. $\endgroup$ Sep 10, 2014 at 11:05
  • $\begingroup$ @TaQ, thank you. I'll check the link which you have provided (but first I need some sleep :-). $\endgroup$ Sep 10, 2014 at 11:13
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    $\begingroup$ @Włodzimierz From sciencedirect.com/science/article/pii/S0893965904817452 `The W*-convexity and normal structure in banach spaces´, Applied Mathematics Letters, 17 (12) 2004, pp. 1381–1386, Def. 1: A bounded, convex subset $K$ of a Banach space $X$ is said to have normal structure if every convex subset $H$ of $K$ that contains more than one point contains a point $x_0\in H$ such that $\sup\,\{\,\|\,x_0-y\,\|:y\in H\,\}<\sup\,\{\,\|\,x-y\,\|:x,y\in H\,\}$ . A Banach space $X$ is said to have normal structure if every bounded, convex subset of $X$ has normal structure. $\endgroup$
    – TaQ
    Sep 10, 2014 at 15:00

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An extension was published in the Taiwanese journal of math in 2009: journal.taiwanmathsoc.org.tw/index.php/tjm/article/download/507/383

I assume (but am not certain) that the original theorem is proved there also

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  • $\begingroup$ Thank You. Your reference helped to get the ideas of the proof of the Brodskii-Milman theorem. $\endgroup$ Sep 10, 2014 at 11:06
  • $\begingroup$ Link rot has struck and the link in this answer no longer works (2021). It's volume 13, issue 2A for anyone searching, and the paper is called Weak and weak$^*$ topologies and Brodskii-Milman's theorem on hyperspaces by Thakyin Hu and Jui-Chi Huang. A currently functioning link to the volume is: projecteuclid.org/journals/taiwanese-journal-of-mathematics/… $\endgroup$
    – postmortes
    Mar 9, 2021 at 9:22

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