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1. CMB Online first
Orbits of Geometric Descent We prove that quasiconvex functions always admit descent trajectories
bypassing all non-minimizing critical points.
Keywords:differential inclusion, quasiconvex function, self-contracted curve, sweeping process Categories:34A60, 49J99 |
2. CMB 2012 (vol 57 pp. 178)
Quasiconvexity and Density Topology We prove that if $f:\mathbb{R}^{N}\rightarrow \overline{\mathbb{R}}$ is
quasiconvex and $U\subset \mathbb{R}^{N}$ is open in the density topology, then
$\sup_{U}f=\operatorname{ess\,sup}_{U}f,$ while
$\inf_{U}f=\operatorname{ess\,inf}_{U}f$
if and only if the equality holds when $U=\mathbb{R}^{N}.$ The first (second)
property is typical of lsc (usc) functions and, even when $U$ is an ordinary
open subset, there seems to be no record that they both hold for all
quasiconvex functions.
This property ensures that the pointwise extrema of $f$ on any nonempty
density open subset can be arbitrarily closely approximated by values of $f$
achieved on ``large'' subsets, which may be of relevance in a variety of
issues. To support this claim, we use it to characterize the common points
of continuity, or approximate continuity, of two quasiconvex functions that
coincide away from a set of measure zero.
Keywords:density topology, quasiconvex function, approximate continuity, point of continuity Categories:52A41, 26B05 |
3. CMB 2012 (vol 57 pp. 25)
Subadditivity Inequalities for Compact Operators Some subadditivity inequalities for matrices and concave functions also hold for Hilbert space operators, but (unfortunately!) with an additional $\varepsilon$ term. It seems not possible to erase this residual term. However, in case of compact operators we show that the $\varepsilon$ term is unnecessary. Further, these inequalities are strict in a certain sense when some natural assumptions are satisfied. The discussion also stresses on matrices and their compressions and several open questions or conjectures are considered, both in the matrix and operator settings.
Keywords:concave or convex function, Hilbert space, unitary orbits, compact operators, compressions, matrix inequalities Categories:47A63, 15A45 |
4. CMB 2011 (vol 55 pp. 697)
Constructions of Uniformly Convex Functions We give precise conditions under which the composition
of a norm with a convex function yields a
uniformly convex function on a Banach space.
Various applications are given to functions of power type.
The results are dualized to study uniform smoothness
and several examples are provided.
Keywords:convex function, uniformly convex function, uniformly smooth function, power type, Fenchel conjugate, composition, norm Categories:52A41, 46G05, 46N10, 49J50, 90C25 |
5. CMB 2005 (vol 48 pp. 283)
Enlarged Inclusion of Subdifferentials This paper studies the integration of inclusion of subdifferentials. Under
various verifiable conditions, we obtain that if two proper lower
semicontinuous functions $f$ and $g$ have the subdifferential of $f$
included in the $\gamma$-enlargement of the subdifferential of $g$, then
the difference of those functions is $ \gamma$-Lipschitz over their
effective domain.
Keywords:subdifferential,, directionally regular function,, approximate convex function,, subdifferentially and directionally stable function Categories:49J52, 46N10, 58C20 |