Is it true that a sphere is a convex set?

Because the thing that I know is that the space enclosed by the sphere is the one convex.

thanks for the help.

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- Nov 27th 2009, 02:23 PMmamenIs a sphere a convex set?
Is it true that a sphere is a convex set?

Because the thing that I know is that the space enclosed by the sphere is the one convex.

thanks for the help. - Nov 27th 2009, 05:10 PMchiph588@
A sphere is convex.

- Nov 27th 2009, 05:58 PMAbu-Khalil
In particular, a sphere encloses itself. Or you are asking bout $\displaystyle \Omega=\left\{(x,y,z)\in\mathbb{R}^3:x^2+y^2+z^2=R ^2\right\}$?

- Nov 28th 2009, 12:03 AMmamen
- Nov 28th 2009, 12:41 AMOpalg
The word "sphere" is used in two different senses. Sometimes it means a solid ball, and sometimes it means the surface of a ball. In mathematics, it's better to use the word "ball" for the solid ball, and to reserve the word "sphere" for the surface of the ball. In that case, the sphere is not convex. Of course, the ball is convex.

A similar thing happens in two dimensions, where the word "circle" is sometimes used to mean a disk, and sometimes (more accurately) it refers to the perimeter of the disk. - Nov 28th 2009, 04:00 AMHallsofIvy
Yes, the "ball" is convex, the "sphere" isn't.

- Nov 28th 2009, 02:13 PMmamen
- Feb 5th 2013, 12:39 AMHAL9000Re: Is a sphere a convex set?
I would like to see how one shows that a sphere $\displaystyle S_r(x_0):=\{x \in E: ||x-x_0||=r\}$ in an arbitrary (not necessarily $\displaystyle \mathbb{R}^n$) normed space $\displaystyle E$ is non-convex.

I would say that this is simply due to the fact that for $\displaystyle x_1,x_2 \in S_r(x_0), t \in [0,1]$ the function $\displaystyle f(t):=||(1-t)(x_1-x_0)+t(x_2-x_0)||$ is non-constant on $\displaystyle (0,1)$. But is it really an adequate argument? There seems to be too much of intuition coming from $\displaystyle \mathbb{R}^n$ in it, don't you find? - Feb 10th 2013, 05:59 AMHallsofIvyRe: Is a sphere a convex set?
- Feb 10th 2013, 06:02 AMHallsofIvyRe: Is a sphere a convex set?
- Feb 24th 2013, 10:09 AMHAL9000Re: Is a sphere a convex set?
I was actually talking about ANY normed space, possibly infinite-dimensional.