description of projections in the convex hull of two surjective linear isometries (carrying a weighted composition operator form) on absolutely continuous function space AC(X, E), where X is a compact subset of
(2.18) It is well known that any convex function is locally absolutely continuous (see, e.g., [59] Proposition 17 of Chapter 5) that is, x2 f (x2) − f (x1) = f (u−)du, x1 0 < x1 < x2 < . (2.19) As the lefthand derivative f (x−) of the convex function...
Lemma . ([, Lemma .] or [, Lemma .]) Let f : [a, b] ⊆ R → R be an n-time differentiable function such that f (n–)(x) for n ∈ N is absolutely continuous on [a, b]. Then the identity b f (x) dx = n– (b – t)...
By Theorem 1.18, f, a proper convex function, is necessarily continuous on ri dom f. As is seen from this theorem, a convex function is continuous in dom f and may have a point of discontinuity only in its boundary. In order to characterize the case in which there is no such ...
14.On level sets of E-convex function and E-quasiconvex function有关E-凸函数和E-拟凸函数的水平集 15.A convex lens is used to concentrate rays of light.凸透镜用于聚集光线。 16.On the Existence of Minimal Convex Generated Set in the Open Set of E~n;E~n中的开集不存在最小凸生成集 ...
convex function is bounded below. We shall say that a function is strongly convex if it is universally measurable, bounded below, and if for all ×∈ K and all (Radon) laws μ on K with barycentre× (29) we have (50.1)f(x)≤∫kf(y)μ(dy) One defines similarly strongly concave,...
Proof. It is easy to see that for any locally absolutely continuous function f :(a,b)→R,we have the identity (2.2) ,x (t−a)f′ (t)dt+,b (t−b)f′ (t)dt=f(x)−,b f(t)dt,for any x∈(a,b)where f′ is the derivative of f which exists a.e. on(a,b) .4 ...
function f (K ,·) :S R,if its surface area measure S (K , ) is absolutely continuous with respect to Le— b esgue m easure S on S ”一 and = f (K ,.) ∈LI( (3) If K is an infi nitely sm ooth b ody w ith p o sitiv e ...
As a consequence, the quadratic function g(h,β)(x) := gh(xH ) + gβ(xV ) is ∆HS -subharmonic on S. Proof. In order to prove the rst claim, let us calculate the HS-Laplacian of the function gh(xH ) by using (i) in Proposition 2.7. We have ∆HS gh(xH ) = h− ...
Let f : [a, b]× [c, d]→ ℝ be an absolutely continuous function such that the partial derivative of order 2 exists and is bounded, i.e., ∂2f(t,s)∂t∂s∞=sup(x,y)∈(a,b)×(c,d)∂2f(t,s)∂t∂s<∞ for all (t, s)∈ [a, b]× [c, d]. Then we...