By G. G. Magaril-Ilyaev, V. M. Tikhomirov

This ebook is an creation to convex research and a few of its functions. It starts off with easy idea, that is defined in the framework of finite-dimensional areas. the single must haves are simple research and easy geometry. the second one bankruptcy offers a few functions of convex research, together with difficulties of linear programming, geometry, and approximation. precise realization is paid to purposes of convex research to Kolmogorov-type inequalities for derivatives of services in a single variable. bankruptcy three collects a few effects on geometry and convex research in infinite-dimensional areas. A complete advent written "for rookies" illustrates the basics of convex research in finite-dimensional areas.

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**Example text**

1 hold and let Λ be a family of partitions with inf τ ∈Λ |τ | = 0. 8) then there exist a sequence (τ k ) ⊂ Λ with |τ k | ↓ 0 and a limit curve u ∈ 2 ACloc ([0, +∞); S ) such that Uτ k (t) σ ∀ t ∈ [0, +∞). 9) σ In particular, if Uτ0k u0 and φ(Uτ0k ) → φ(u0 ) as k → ∞, then u(0+) = u0 and u ∈ GM M (Φ; u0 ), which is therefore a nonempty set. We prove this proposition in the next Section 3. 4 (p-estimates). 3 still hold (with diﬀerent constants) simply replacing 2 with p: thus the p limiting curve belongs to ACloc ([0, +∞); S ).

10b) with respect to the weak Lp -topology σ. e. in Rd so that η = F (v) and therefore ξ ∈ ∂φ(v). 4 The (geodesically) convex case In this section we will consider a notion of convexity along classes of curves in the metric space S : a particular attention is devoted to functionals φ which are convex along the geodesics of the metric space S . Let us ﬁrst introduce the relevant deﬁnitions. 50 Chapter 2. 1 (λ-convexity along curves). A functional φ : S → (−∞, +∞] is called convex on a curve γ : t ∈ [0, 1] → γt ∈ S if φ(γt ) ≤ (1 − t)φ(γ0 ) + tφ(γ1 ) ∀t ∈ [0, 1].

4). ∂ ◦ φ(v) is the subset of elements of minimal (dual) norm in ∂φ(v), which reduces to a single point if the dual norm of B is strictly convex. Notice that |∂φ|(v) = lim sup w→0 φ(v) − φ(v + w) w ≤ lim sup ξ, w w w→0 ≤ ξ ∗ ∀ ξ ∈ ∂φ(v). 34 Chapter 1. 5 that the map v → ∂ ◦ φ(v) ∗ is a weak upper gradient for φ. e. 10) as the solution of a suitable doubly nonlinear diﬀerential inclusion: in the case when S is a reﬂexive Banach space and φ is convex, these kind of evolution equations have been studied in [53, 52]; we refer to these contributions and to [148] for many examples of partial diﬀerential equations which can be studied by this abstract approach.