The Banach space $$\mathcal {P}({}^2X)$$ of 2-homogeneous polynomials on the X can be naturally embedded in $${\mathrm{Lip}_0}(B_X)$$ real-valued Lipschitz functions $$B_X$$ that vanish at 0. We investigate whether is a complemented subspace . This line research considered as polynomial counterpart to classical result by Joram Lindenstrauss, asserting {P}({}^1X)=X^*$$ for every X. Our main asse...

In this note, we consider the equiprime and strongly prime radicals in the near-ring LX of Lipschitz functions over a normed vector space X. We prove that the equiprime radical of LX is trivial, and we also obtain upper and lower bounds on its strongly prime radical in terms of the ideal of bounded Lipschitz functions on X. 2000 AMS Classification: 16Y30.

In a recent paper we have shown that most non-expansive Lipschitz functions (in the sense of Baire’s category) have a maximal Clarke subdifferential. In the present paper, we show that in a separable Banach space the set of non-expansive Lipschitz functions with a maximal Clarke subdifferential is not only of generic, but also staunch. 1991 Mathematics Subject Classification: Primary 49J52.

Rademacher and Gaussian complexities are successfully used in learning theory for measuring the capacity of the class of functions to be learned. One of the most important properties for these complexities is their Lipschitz property: a composition of a class of functions with a fixed Lipschitz function may increase its complexity by at most twice the Lipschitz constant. The proof of this prope...

We consider finite sample properties of the regularized high-dimensional Cox regression via lasso. Existing literature focuses on linear models or generalized linear models with Lipschitz loss functions, where the empirical risk functions are the summations of independent and identically distributed (iid) losses. The summands in the negative log partial likelihood function for censored survival...

Motivated by applications to (directionally) Lipschitz functions, we provide a general result on the almost everywhere Gâteaux differentiability of real-valued functions on separable Banach spaces, when the function is monotone with respect to an ordering induced by a convex cone with nonempty interior. This seemingly arduous restriction is useful, since it covers the case of directionally Lips...

In this paper we address the problem of characterizing the in nitesimal properties of functions which are non-increasing along all the trajectories of a di erential inclusion. In particular, we extend the condition based on the proximal gradient to the case of semicontinuous functions and Lipschitz continuous di erential inclusions. Moreover, we show that the same criterion applies also in the ...

A planar set D will be called a lip domain if it is Lipschitz, open, bounded, connected, and given by (1) D = {(x1, x2) : f1(x1) < x2 < f2(x1)}, where f1, f2 are Lipschitz functions with constant 1. The assumption that D is a Lipschitz domain puts an extra constraint on the functions fk; we discuss this issue in greater detail later in this section. Let μ2 denote the second eigenvalue for the L...

In this work we prove a new Lp holomorphic extension result for functions defined on product Lipschitz surfaces with small Lipschitz constants in two complex variables. We define biparameter and partial Cauchy integral operators that play the role of boundary values for holomorphic functions on product Lipschitz domain. In the spirit of the application of David-Journé-Semmes [DJS85] and Christ’...