How do you prove strictly convex?

How do you prove strictly convex?

(1) The function is strictly convex if the inequality is always strict, i.e. if x = y implies that θf ( x) + (1 − θ)f ( y) > f (θ x + (1 − θ) y). (2) A concave function is a function f such that −f is convex. Linear functions are convex, but not strictly convex.

What is strictly convex set?

A set C is strictly convex if every point on the line segment connecting x and y other than the endpoints is inside the interior of C. A set C is absolutely convex if it is convex and balanced.

How do you prove a ray is a convex set?

Ray is Convex

1. Let (S,⪯) be an ordered set.
2. Let I be a ray, either open or closed.
3. Then I is convex in S.
4. Without loss of generality, suppose that U is an upward-pointing ray.
5. according to whether U is open or closed.
6. Thus I is convex.

Is X1 * X2 convex?

If g : Rn → R is a convex function, then the set X = {x : g(x) ≤ 0 is a convex set. Proposition 8. If the sets X1 and X2 are convex, then the set X = X1 ∩ X2 is convex as well.

Can an open set be convex?

Note: open convex sets have no extreme points, as for any x ∈ X there would be a small ball Br(x) ⊂ X, in which case any d is a direction, at any x.

What is the difference between convex and strictly convex?

Geometrically, convexity means that the line segment between two points on the graph of f lies on or above the graph itself. Strict convexity means that the line segment lies strictly above the graph of f, except at the segment endpoints.

What is convex set and non-convex set?

Definition. A set X ∈ IRn is convex if ∀x1,x2 ∈ X, ∀λ ∈ [0, 1], λx1 + (1 − λ)x2 ∈ X. A set is convex if, given any two points in the set, the line segment connecting them lies entirely inside the set. Convex Sets. Non-Convex Sets.

How is a convex set defined?

A convex set is a set of points such that, given any two points A, B in that set, the line AB joining them lies entirely within that set. A convex set; no line can be drawn connecting two points that does not remain completely inside the set.

Is e x strictly convex?

The function ex is differentiable, and its second derivative is ex > 0, so that it is (strictly) convex. Hence by a result in the text the set of points above its graph, {(x, y): y ≥ ex} is convex. Convex: see the following figure.

Is x3 convex?

The function x3 has second derivative 6x; thus it is convex on the set where x ≥ 0 and concave on the set where x ≤ 0.

Can you maximize a convex function?

Given a convex function f:Rn→[0,∞), the objective is to find the farthest point in the level set {x∈Rn∣f(x)≤1} (Assuming that such set is non empty, and closed and compact), i.e. maximizex∈Rn||x||2subject tof(x)≤1.

Is a line a convex set?

A set is convex if it includes all convex combinations of points in the set. Or in other words, if it contains all line segment joining any two points in the set. Thus, a line is a convex set.

Is a sphere a convex set?

A sphere is convex. But a sphere has a space inside where in that space is not a part of the sphere.

What is a convex math?

In mathematics, a real-valued function defined on an n-dimensional interval is called convex (or convex downward or concave upward) if the line segment between any two points on the graph of the function lies above or on the graph.

What is convex function?

Convex Function. A convex function is a continuous function whose value at the midpoint of every interval in its domain does not exceed the arithmetic mean of its values at the ends of the interval. (Rudin 1976, p. 101; cf.