We consider spectral functions f ◦ λ, where f is any permutation-invariant mapping from Cn to R, and λ is the eigenvalue map from the set of n × n complex matrices to Cn , ordering the eigenvalues lexicographically. For example, if f is the function “maximum real part”, then f ◦ λ is the spectral abscissa, while if f is “maximum modulus”, then f ◦ λ is the spectral radius. Both these spectral f...

After considering a variant of the generalized mean value inequality of quasinearly subharmonic functions, we consider certain invariance properties of quasinearly subharmonic functions. Kojić has shown that in the plane case both the class of quasinearly subharmonic functions and the class of regularly oscillating functions are invariant under conformal mappings. We give partial generalization...

Steepest descent drives both theory and practice of nonsmooth optimization. We study slight relaxations of two influential notions of steepest descent curves — curves of maximal slope and solutions to evolution equations. In particular, we provide a simple proof showing that lower-semicontinuous functions that are locally Lipschitz continuous on their domains — functions playing a central role ...

Lipschitz quotient mappings and uniform quotient mappings between Banach spaces are introduced and investigated. In particular, it is shown that if a Banach space is a uniform quotient of L p , 1 < p < ∞, then it is isomorphic to a linear quotient of L p. For the purpose of studying quotient mappings and also for their own interest, new concepts related to approximating general Lipschitz functi...

where g satis…es a Lipschitz condition with the constant K: In (1.1) and (1.2) the constant 14 is best possible in the sense that it cannot be replaced by a smaller quantity. The same applies for the constant 1 12 in (1.3). The interested reader may …nd many other inequalities involving Lipschitzian functions in the papers [1]-[11], [13]-[20] and in the references therein. In this paper we esta...

We study the relationships between Gateaux, Weak Hadamard and Fréchet differentiability and their bornologies for Lipschitz and for convex functions. AMS Subject Classification. Primary: 46A17, 46G05, 58C20. Secondary: 46B20.

Subdifferentials of convex functions and some regular functions f are expressed in terms of limiting gradients at points in a given dense subset of dorn Vf.

In this paper we extend the notion of a Lebesgue-null set to a notion which is valid in any completely metrizable Abelian topological group. We then use this deenition to introduce and study the class of essentially smooth functions. These are, roughly speaking, those Lipschitz functions which are smooth (in each direction) almost everywhere.

This paper provides a version of the transversality theorem for a class of Lipschitz functions of the form f : R × C → Rn where C is a convex subset of a normed vector space Z indexing the parameters in the problem. The set C may be infinite-dimensional, and the notion of generic used is the measure-theoretic notion of prevalence introduced by Hunt, Sauer and Yorke (1992) and Christensen (1974)...

For a metric space X, we study the space D∞(X) of bounded functions on X whose infinitesimal Lipschitz constant is uniformly bounded. D ∞(X) is compared with the space LIP∞(X) of bounded Lipschitz functions on X, in terms of different properties regarding the geometry of X. We also obtain a Banach-Stone theorem in this context. In the case of a metric measure space, we also compare D∞(X) with t...