Bi-Lipschitz concordance implies bi-Lipschitz isotopy

We modify the proof of an earlier result of ours to deforming topological, bi-Lipschitz, and quasiconformal embeddings of an open subsetU ofR ⁿ which now are of small uniform distance from the inclusion map. As an application we show that two bi-Lipschitz homeomorphismsf ₀,f ₁:R ⁿRⁿ are bi-Lipschitz isotopic if and only ifd(f ₀,f ₁)<.Research supported in part by a grant from the Institut Mittag-Leffler.     

Semianalytic and subanalytic sets

Dedicated to Stanislaw Łojasiewicz for his sixtieth birthday.     

Smooth points ofp-adic subanalytic sets

Ap-adic subanalytic set shares with a real subanalytic set the fundamental property that its singular locus is itself subanalytic. Furthermore, given ap-adic subanalytic function ? with domain contained in ? _p ^m , there is an integerL such that for any pointx ₀ ∈ ? _p ^m in a neighborhood of whichf is defined,f has a Taylor approximation up to orderL atx ₀ if, and only if, ? is analytic aroundx ₀. These results extend to thep-adic fields real variables theorems by M. Tamm [21].     

On regular tangent cones

The paper contains a sufficient condition for an intersection of regular tangent cones to be a tangent cone. Regular tangent cones and tents for sets given by locally Lipschitz functions are constructed. The cones are described in terms of generalized K-derivatives.     

Equivariant uniformization theorem for subanalytic sets

In this paper we prove an equivariant version of the uniformization theorem for closed subanalytic sets: Let G be a Lie group and let M be a proper real analytic G-manifold. Let X be a closed subanalytic G-invariant subset of M. We show that there exist a proper real analytic G-manifold N of the same dimension as X and a proper real analytic G-equivariant map such that .      

Nearly optimal meshes in subanalytic sets

We prove that any fat, subanalytic compact subset of $\mathbb R^N$ possesses a nearly optimal (polynomial) admissible mesh. It is related to particular results that have recently appeared in the literature for very special (globally semianalytic) sets like N-dimensional polynomial or analytic graph domains or polynomial and analytic polyhedrons. (Here a good source of references is the recent paper (Piazzon and Vianello, East J Approx 16(4):389?C398, 2010).) We also show that an infinitely differentiable map f from a compact set Q in $\mathbb R^N$ onto a Markov compact set K in $\mathbb C^l$ (l????N) transforms a (weakly) admissible mesh in Q onto a (weakly) admissible mesh in K, which extends a result of Piazzon and Vianello (East J Approx 16(4):389?C398, 2010) for analytic maps in case Q is a subset of $\mathbb R^N$ . Versions for $\mathcal C^k$ maps with sufficiently large k are also given.     

On the normal cycles of subanalytic sets

We present a very short and complete proof of the existence of the normal cycle of a subanalytic set. The approach is a blend of Morse theory and geometric integration theory and relies heavily on techniques from o-minimal geometry.     

Isotone tangent cones and nonsmooth optimization

A simple but quite general method of formulating necessary optimality conditions for an abstract mathematical program is presented. This method centers around tangent cones which are isotone with respect to inclusion. The optimality conditions derived by this method generalize standard necessary conditions of differentiate, convex, and nonsmooth mathematical programming.     

On the structure of p-adic subanalytic functions and sets

We prove a p-adic version of the Lion?CRolin preparation theorem. As a consequence, we obtain a cell decomposition theorem, which can be viewed as an extension of Denef??s cell decomposition theorem to the p-adic analytic case. Using these theorems, we give shorter and more explicit proofs of some of the results by Denef and van den Dries on p-adic subanalytic sets.     

Lipschitz constant for bi-Lipschitz automorphism on Moran-like sets

This paper proves that if F⊂R¹ is a complete set equipped with some suitable Moran-like structure, then there is a constant c₀>1 such that for any bi-Lipschitz bijection ,

     

An infinite class of convex tangent cones

Since the early 1970's, there have been many papers devoted to tangent cones and their applications to optimization. Much of the debate over which tangent cone is best has centered on the properties of Clarke's tangent cone and whether other cones have these properties. In this paper, it is shown that there are an infinite number of tangent cones with some of the nicest properties of Clarke's cone. These properties are convexity, multiple characterizations, and proximal normal formulas. The nature of these cones indicates that the two extremes of this family of cones, the cone of Clarke and the B-tangent cone or the cone of Michel and Penot, warrant further study. The relationship between these new cones and the differentiability of functions is also considered.     

Complements of subanalytic sets and existential formulas for analytic functions

We show that the complement of a subanalytic set defined by real analytic functions from any subalgebra closed under differentiation is a subanalytic set defined by the functions from the same subalgebra. This result has an equivalent formulation in logic: Consider an expression built from functions as above using equalities and inequalities as well as existential and universal quantifiers. Such an expression is equivalent to an existential expression involving functions from the same class, provided that the variables approach neither infinity nor the boundary of the domain. Oblatum 17-VIII-1995     

Non‐archimedean stratifications of tangent cones

We study the impact of a kind of non‐archimedean stratifications (t‐stratifications) on tangent cones of definable sets in real closed fields. We prove that such stratifications induce stratifications of the same nature on the tangent cone of a definable set at a fixed point. As a consequence, the archimedean counterpart of a t‐stratification is shown to induce Whitney stratifications on the tangent cones of a semi‐algebraic set. Extensions of these results are proposed for real closed fields with further structure.