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Schatten $p$ norm is convex when $p\geq1$ holds and if $p\in(0,1)$ it is quasinorm.

  1. If $p\in(0,1)$ then is Schatten $p$ norm quasi convex? I am interested in definition of quasi convexity here https://dl.acm.org/citation.cfm?id=1716350 and here http://www.numdam.org/item/COCV_2008__14_4_795_0.

  2. Is there a general criterion when quasinorms are quasiconvex?

Is there a good reference on quasi convex matrix norms?

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No, quasinorms are in general not quasiconvex. (Well, this is true if quasiconve means that the levelsets of the function are convex; other defintions of quasiconvexity may exist…)

By positive homogeneity, a (quasi)norm is completely described by one of its levelsets. If this levelset is convex (and fulfills some other technical conditions) the function is actually a norm, but the property that separates norm from quasinorms is the convexity of the levelsets.

For the specific question: The Schatten $p$-quasinorm of a diagonal matrix with positive entries on the diagonal is the $p$-quasinorm of the diagonal entries and thus, for $0<p<1$, the Schatten quasinorm is not quasiconvex.

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  • $\begingroup$ Modifieded with reference of quasiconvexity of interest. $\endgroup$
    – Turbo
    Commented Sep 24, 2018 at 10:26
  • $\begingroup$ Now I am confused. One link defines quasiconvexity in the way I used it, the other defines quasiconvexity in another way which is applicable to functions of a matrix argument. Which one are you interested in? $\endgroup$
    – Dirk
    Commented Sep 24, 2018 at 11:14
  • $\begingroup$ I am interested in matrix schatten norm sorry for being unclear. $\endgroup$
    – Turbo
    Commented Sep 24, 2018 at 11:36
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    $\begingroup$ I got that, but which notion of quasiconvexity are you interested in? There are at least two different meanings (and you linked both of them). $\endgroup$
    – Dirk
    Commented Sep 24, 2018 at 11:39
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    $\begingroup$ I am interested in the convex programming case. $\endgroup$
    – Turbo
    Commented Sep 24, 2018 at 12:33

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