Attainment and (Sub)differentiability of the Inﬁmal/Supremal Convolution
CIBULKA, R. Attainment and (Sub)differentiability of the Inﬁmal/Supremal Convolution. Limoges, France, 2012.
|Anglický název:||Attainment and (Sub)differentiability of the Inﬁmal/Supremal Convolution|
|Autoři:||Ing. Radek Cibulka Ph.D.|
|Abstrakt EN:||S. Fitzpatrick showed that there is a close relationship between the results on a distance function and on a farthest distance function. But one has to be cautious, because some properties of the farthest distance function have no counterpart in the distance function. Comparing results by S. Dutta, this connection emerges again. Recently X. Wang established the symmetry concerning properties of the Moreau envelope (infimal convolution) and the Klee envelope (supremal convolution), as well as of the corresponding proximal mapping and the farthest-point mapping, in Euclidean spaces. Continuing in the same vein, we present several statements on the attainment and (sub)differentiability of the infimal (supremal) convolution of a fairly general function f and a function of the norm, e.g. the square of the norm. First, we study the subdifferential of the infimal (supremal) convolution in relation to the subdifferential of the norm of the underlying Banach space. Second, if the norm is simultaneously LUR and Gateaux (Fréchet) smooth, we show that the infimal (supremal) convolution of f and the square of the norm is generically strongly attained and hence it is Gateaux (Fréchet) differentiable.|