Explicit formulas for C1,1 and Cconv1,ω extensions of 1-jets in Hilbert and superreflexive spaces

Kirszbraun’s Theorem via an Explicit Formula

“…(3) If c > 0, then G is strongly biLipschitz with SBilip(G) ≥ 2 c+M min{1, cM }. (4) For c = −M we recover Wells's condition W 1,1 considered in [24,18,3]. For c = 0, (SCW 1,1 ) is just condition (CW 1,1 ) of [2,3].…”

“…Proof. The necessity of the condition (SCW is convex and of class C 1,1 (X) with Lip(∇ F ) ≤ M − c and ( F , ∇ F ) = ( f , G) on E. If we consider the function F := F + c 2 · 2 , Lemma 11 says that (F, ∇F ) satisfies condition (SCW 1,1 ) with constants (c, M ) on X (because ( F , ∇ F ) satisfies (CW 1,1 ) with constant M − c on X) and (F, ∇F ) = (f, G) on E. It is obvious that F − c 2 · 2 is convex on X and, by Remark 10 (2), Lip(∇F ) ≤ M. Finally, if H is a function of class C 1,1 (X) such that (H, ∇H) = (f, G) on E, Lip(∇H) ≤ M and H := H − c 2 · 2 is convex, then it is easy to see (using the same calculations as in the proof of Proposition 12 (2)) that Lip(∇ H) ≤ M − c, and obviously ( H, ∇ H) = ( f , G) on E. We thus have from [3,Theorem 2.4] that H ≤ F on X, and therefore H ≤ F on X.…”

“…(4) For c = −M we recover Wells's condition W 1,1 considered in [24,18,3]. For c = 0, (SCW 1,1 ) is just condition (CW 1,1 ) of [2,3]. For c ∈ (0, M ] we have what can be called a C 1,1 strongly convex 1-jet, which in the extreme case c = M becomes the restriction of a parabola to E.…”

Extensions of convex functions with prescribed subdifferentials

“…Very recently, a related problem has been solved for convex functions of the classes C 1 (R n ) and C 1,ω (X) (for a Hilbert space X) in the situation where the mapping G is single-valued and one additionally requires that the extension F be of class C 1 (R n ) (which amounts to asking that ∂F (x) be a singleton for every x ∈ R n ) or of class C 1,ω (X); see [1,2,3]. A solution to a similar problem for general (not necessarily convex) functions was given in [11,Theorem 5], characterizing the pairs f : E → R, G : E ⇒ R n with f continuous and G upper semicontinuous and nonempty, compact and convexvalued which admit a (generally nonconvex) extension F of f whose Fréchet subdifferential is upper semicontinuous on R n and extends G from E.…”

“…However, in the case that E is unbounded (and even if E is closed), condition (P CS) cannot replace (SCS), as the following example shows. 1], and θ affine on [1, +∞). Let E = {(0, 0)} ∪ {(x, y) ∈ R 2 : |x| ≥ min{1, e y }}, and define f and G on E by…”