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The function () = has ″ = >, so f is a convex function. It is also strongly convex (and hence strictly convex too), with strong convexity constant 2. The function () = has ″ =, so f is a convex function. It is strictly convex, even though the second derivative is not strictly positive at all points.
( i.e. be holomorphically convex.) such that f cannot be extended to any neighbourhood of x. i.e., let : be a holomorphic map, if every point has a neighborhood U such that () admits a -plurisubharmonic exhaustion function (weakly 1-complete [55]), in this situation, we call that X is locally pseudoconvex (or locally Stein) over Y. As an old ...
The Banach space (X, ǁ ⋅ ǁ) is a strictly convex space (i.e., the boundary of the unit ball B contains no line segments) if and only if δ(2) = 1, i.e., if only antipodal points (of the form x and y = −x) of the unit sphere can have distance equal to 2. When X is uniformly convex, it admits an equivalent norm with power type modulus of ...
This follows from the fact that the logarithm is monotone implying that the superlevel sets of this function are convex. [1] Every concave function that is nonnegative on its domain is log-concave. However, the reverse does not necessarily hold. An example is the Gaussian function f(x) = exp(−x 2 /2) which is log-concave since log f(x) = −x ...
We are given convex function to optimize over a convex set , and given some norm ‖ ‖ on . We are also given differentiable convex function h : R n → R {\displaystyle h\colon \mathbb {R} ^{n}\to \mathbb {R} } , α {\displaystyle \alpha } - strongly convex with respect to the given norm.
Strictly convex function, a function having the line between any two points above its graph; Strictly convex polygon, a polygon enclosing a strictly convex set of points; Strictly convex set, a set whose interior contains the line between any two points; Strictly convex space, a normed vector space for which the closed unit ball is a strictly ...
Strictly logarithmically convex if is strictly convex. Here we interpret as . Explicitly, f is logarithmically convex if and only if, for all x 1, x 2 ∈ X and all t ∈ [0, 1], the two following equivalent conditions hold:
Let : be a continuously-differentiable, strictly convex function defined on a convex set. The Bregman distance associated with F for points p , q ∈ Ω {\displaystyle p,q\in \Omega } is the difference between the value of F at point p and the value of the first-order Taylor expansion of F around point q evaluated at point p :