Displaying 221 – 240 of 389

Showing per page

On the points of non-differentiability of convex functions

David Pavlica (2004)

Commentationes Mathematicae Universitatis Carolinae

We characterize sets of non-differentiability points of convex functions on n . This completes (in n ) the result by Zajíček [2] which gives a characterization of the magnitude of these sets.

On the quasiconvex exposed points

Kewei Zhang (2001)

ESAIM: Control, Optimisation and Calculus of Variations

The notion of quasiconvex exposed points is introduced for compact sets of matrices, motivated from the variational approach to material microstructures. We apply the notion to give geometric descriptions of the quasiconvex extreme points for a compact set. A weak version of Straszewicz type density theorem in convex analysis is established for quasiconvex extreme points. Some examples are examined by using known explicit quasiconvex functions.

On the quasiconvex exposed points

Kewei Zhang (2010)

ESAIM: Control, Optimisation and Calculus of Variations

The notion of quasiconvex exposed points is introduced for compact sets of matrices, motivated from the variational approach to material microstructures. We apply the notion to give geometric descriptions of the quasiconvex extreme points for a compact set. A weak version of Straszewicz type density theorem in convex analysis is established for quasiconvex extreme points. Some examples are examined by using known explicit quasiconvex functions.

On the reduction of pairs of bounded closed convex sets

J. Grzybowski, D. Pallaschke, R. Urbański (2008)

Studia Mathematica

Let X be a Hausdorff topological vector space. For nonempty bounded closed convex sets A,B,C,D ⊂ X we denote by A ∔ B the closure of the algebraic sum A + B, and call the pairs (A,B) and (C,D) equivalent if A ∔ D = B ∔ C. We prove two main theorems on reduction of equivalent pairs. The first theorem implies that, in a finite-dimensional space, a pair of nonempty compact convex sets with a piecewise smooth boundary and parallel tangent spaces at some boundary points is not minimal. The second theorem...

On the Representation of Effective Energy Densities

Christopher J. Larsen (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We consider the question raised in [1] of whether relaxed energy densities involving both bulk and surface energies can be written as a sum of two functions, one depending on the net gradient of admissible functions, and the other on net singular part. We show that, in general, they cannot. In particular, if the bulk density is quasiconvex but not convex, there exists a convex and homogeneous of degree 1 function of the jump such that there is no such representation.

On the Rockafellar theorem for Φ γ ( · , · ) -monotone multifunctions

S. Rolewicz (2006)

Studia Mathematica

Let X be an arbitrary set, and γ: X × X → ℝ any function. Let Φ be a family of real-valued functions defined on X. Let Γ : X 2 Φ be a cyclic Φ γ ( · , · ) -monotone multifunction with non-empty values. It is shown that the following generalization of the Rockafellar theorem holds. There is a function f: X → ℝ such that Γ is contained in the Φ γ ( · , · ) -subdifferential of f, Γ ( x ) Φ γ ( · , · ) f | x .

Currently displaying 221 – 240 of 389