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?



Quasiconvexity property of quasinorms
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.


