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# Strong pseudoconvexity in Banach spaces

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### Abstract

Having been unclear how to define strong (or strict) pseudoconvexity in the infinite-dimensional and non-smooth boundary context, we take a look at the available literature on strong pseudoconvexity, focusing first in eliminating the need of two degress of smoothness e.g. via distributions. We pass to the infinite-dimensional setting by first seeing a uniform notion of strict pseudoconvexity. Since $$2$$-uniformly PL-convex Banach spaces play an important role as examples with uniformly pseudoconvex unit ball, we briefly show that $$r$$-uniformly convex spaces are $$r$$-uniformly PL-convex, and prove a number of characterizations of $$r$$-uniform PL-convexity.

### Most cited references9

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### Sharp uniform convexity and smoothness inequalities for trace norms

(1994)
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### Logarithmic Sobolev inequalities and hypercontractive estimates on the circle

(1980)
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### Domaines pseudoconvexes infinis et la m\'etrique riemannienne dans un espace projectif

(1964)
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### Author and article information

###### Journal
2017-01-13
###### Article
1701.03823

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

###### Custom metadata
32T15, 32C15, 46B25, 32A70
17 pages
math.CV

Analysis