Question

Real Analysis II
(Please do this only if you are sure)
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I am also providing the convex set definition

And key details from my book which surely helps

11. Show that K is a convex set by directly applying the definition. Sketch K in the cases n= 1, 2, 3. is a basis for E. This is the n-parallelepiped spanned by vı, vertex 1% with 0 as a Definition. Let K E". Then K is a convex set if the line segment joining any two points of K is contained in K (Figure 1.6). question is actually convex. To show that a set K is convex directly from the definition, we must verify that for every x1, X2 e K and t e [0, 1], the point x - txi + (1 - t)x2 also belongs to K. In the definition, we assumed that X,丈X2 . But if x,-x2, it is trivial that x e K, since x = x,-X2 Example 2. Any closed half-space is a convex set. Let H - [x:z-x 2 c), z 0. Let X1, X2eH and x =x,t(1-)x2+ where te [0, 1]. Then (1-1)z . x2 (1 -r)('. Consequently, This shows that x e H. Therefore H is a convex set. Similarly, any hyperplane is a convex set (Problem 12) and any open half-space is a convex set. EXAMPLE 3. Let be an open spherical n-ball, namely, U = {x : |x-xol 0, To show that U is a convex set, we proceed as in Example 2. Let X1, X2 e U and x = x, t (1-)x2, where țe [0, 1]. Then xo = t(x,-xo) + (1-)(x2-Xo), Hence x EU

Answer 1

Hence where is a Convex Set ven 1/2

where λ>0 be and Scala Y tuna 04di) So, k is a Convex Set 리2