On the first integral conjecture of Rene Thom

More that half a century ago R. Thom asserted in an unpublished manuscript that, generically, vector fields on compact connected smooth manifolds without boundary can admit only trivial continuous first integrals. Though somehow unprecise for what concerns the interpretation of the word \textquotedblleft generically\textquotedblright, this statement is ostensibly true and is nowadays commonly accepted. On the other hand, the (few) known formal proofs of Thom's conjecture are all relying to the classical Sard theorem and are thus requiring the technical assumption that first integrals should be of class $C^{k}$ with $k\geq d,$ where $d$ is the dimension of the manifold. In this work, using a recent nonsmooth extension of Sard theorem we establish the validity of Thom's conjecture for locally Lipschitz first integrals, interpreting genericity in the $C^{1}$ sense

Locally symmetric submanifolds lift to spectral manifolds

In this work we prove that every locally symmetric smooth submanifold gives rise to a naturally defined smooth submanifold of the space of symmetric matrices, called spectral manifold, consisting of all matrices whose ordered vector of eigenvalues belongs to the locally symmetric manifold. We also present an explicit formula for the dimension of the spectral manifold in terms of the dimension and the intrinsic properties of the locally symmetric manifold

Sweeping by a tame process

We show that any semi-algebraic sweeping process admits piecewise absolutely continuous solutions, and any such bounded trajectory must have finite length. Analogous results hold more generally for sweeping processes definable in o-minimal structures. This extends previous work on (sub)gradient dynamical systems beyond monotone sweeping sets.Comment: 18 page

Linear structure of functions with maximal Clarke subdifferential

It is hereby established that the set of Lipschitz functions $f:\mathcal{U}\rightarrow \mathbb{R}$ ($\mathcal{U}$ nonempty open subset of $\ell_{d}^{1}$) with maximal Clarke subdifferential contains a linear subspace of uncountable dimension (in particular, an isometric copy of $\ell^{\infty}(\mathbb{N})$). This result goes in the line of a previous result of Borwein-Wang. However, while the latter was based on Baire category theorem, our current approach is constructive and is not linked to the uniform convergence. In particular we establish lineability (and spaceability for the Lipschitz norm) of the above set inside the set of all Lipschitz continuous functions

The slope robustly determines convex functions

We show that the deviation between the slopes of two convex functions controls the deviation between the functions themselves. This result reveals that the slope -- a one dimensional construct -- robustly determines convex functions, up to a constant of integration

Explicit formulas for C1,1C^{1,1} Glaeser-Whitney extensions of 1-fields in Hilbert spaces

We give a simple alternative proof for the $C^{1,1}$--convex extension problem which has been introduced and studied by D. Azagra and C. Mudarra [2]. As an application, we obtain an easy constructive proof for the Glaeser-Whitney problem of $C^{1,1}$ extensions on a Hilbert space. In both cases we provide explicit formulae for the extensions. For the Gleaser-Whitney problem the obtained extension is almost minimal, that is, minimal up to a factor $\frac{1+\sqrt{3}}{2}$ in the sense of Le Gruyer [15]