Question

Prove sphere is convex set

Answer 1

I have to show that the unit sphere represented by is convex.

A set is said to be convex when sx+(1−s)y∈M, where x,y∈M and s∈(0,1)

I've read on wikipedia that this can be proven over the triangle inequality, but I think it can be solved in another way? Would this be enough as proof:

For the unit sphere, we have to prove that 0≤sx+(1−s)y≤1 (because ||x||≤1therefore, 0≤x,y≤1). Seeing as the maximal value that x and y can take are 1, the maximum the equation can achieve is 1 (when s=1,x=1 or s=0,y=1). The same can be shown for the minimum 0, therefore it is really between 0 and 1. Finished?

Many thanks in advance!

Answer 2

Answer:

A set is called a convex set when it satisfy the condition sx + (1-s)yeM and s lies between 0 and 1.

All set of sphere values satisfies this condition and therefore sphere is called a convex set. Check the sphere values and you will get a convex set.

Step-by-step explanation:

Consider the norm of sx+(1−s)y. Let x, y belongs to B'x and s belongs to [0, 1], then by triangle inequality the equation forms is

||sx+(1−s)y||≤||sx||+||(1−s)y||=s||x||+(1−s)||y||≤s+(1−s)=1.

Therefore, sx+(1−s)y belongs to B'x is said to be convex. The convex of any type of sphere is proved by the triangle inequality law. The value of s must lies between 0 and 1.

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