16,193 research outputs found
On sets of non-differentiability of Lipschitz and convex functions
summary:We observe that each set from the system (or even ) is -null; consequently, the version of Rademacherâs theorem (on GĂąteaux differentiability of Lipschitz functions on separable Banach spaces) proved by D. Preiss and the author is stronger than that proved by D. Preiss and J. Lindenstrauss. Further, we show that the set of non-differentiability points of a convex function on is -strongly lower porous. A discussion concerning sets of FrĂ©chet non-differentiability points of continuous convex functions on a separable Hilbert space is also presented
Area theorem and smoothness of compact Cauchy horizons
We obtain an improved version of the area theorem for not necessarily
differentiable horizons which, in conjunction with a recent result on the
completeness of generators, allows us to prove that under the null energy
condition every compactly generated Cauchy horizon is smooth and compact. We
explore the consequences of this result for time machines, topology change,
black holes and cosmic censorship. For instance, it is shown that compact
Cauchy horizons cannot form in a non-empty spacetime which satisfies the stable
dominant energy condition wherever there is some source content.Comment: 44 pages. v2: added Sect. 2.4 on the propagation of singularities and
a second version of the area theorem (Theor. 14) which quantifies the area
increase due to the jump se
Nearest points and delta convex functions in Banach spaces
Given a closed set in a Banach space , a point
is said to have a nearest point in if there exists such that
, where is the distance of from . We shortly
survey the problem of studying how large is the set of points in which have
nearest points in . We then discuss the topic of delta-convex functions and
how it is related to finding nearest points.Comment: To appear in Bull. Aust. Math. So
Limited operators and differentiability
We characterize the limited operators by differentiability of convex
continuous functions. Given Banach spaces and and a linear continuous
operator , we prove that is a limited operator if
and only if, for every convex continuous function and
every point , is Fr\'echet differentiable at
whenever is G\^ateaux differentiable at
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