We denote by $K$ the set of triplets $(\alpha,\beta,\gamma)$ of $\mathbf{R}^3$ consisting of three non-negative real numbers such that $\alpha+\beta+\gamma=1$. If $(a,b,c)\in\mathbf{C}^3$, we denote by $\widehat{abc}$ the filled triangle defined by $\widehat{abc} = \{\alpha a + \beta b + \gamma c / (\alpha,\beta,\gamma)\in K\}$. a) Prove that $K$ is a compact subset of $\mathbf{R}^3$ for its usual topology. b) Prove that $K$ is convex, that is, for every real $t\in[0,1]$ and every pair $(u,v)$ of elements of $K$, $tu+(1-t)v$ belongs to $K$. c) Establish that, if $(a,b,c)\in\mathbf{C}^3$, $\widehat{abc}$ is a compact convex subset of $\mathbf{C}$ equipped with its usual topology. d) With the same notation prove the existence of: $$\delta(\widehat{abc}) = \max\left\{|z'-z| / (z,z')\in\widehat{abc}^2\right\}$$
We denote by $K$ the set of triplets $(\alpha,\beta,\gamma)$ of $\mathbf{R}^3$ consisting of three non-negative real numbers such that $\alpha+\beta+\gamma=1$. If $(a,b,c)\in\mathbf{C}^3$, we denote by $\widehat{abc}$ the filled triangle defined by $\widehat{abc} = \{\alpha a + \beta b + \gamma c / (\alpha,\beta,\gamma)\in K\}$.
a) Prove that $K$ is a compact subset of $\mathbf{R}^3$ for its usual topology.
b) Prove that $K$ is convex, that is, for every real $t\in[0,1]$ and every pair $(u,v)$ of elements of $K$, $tu+(1-t)v$ belongs to $K$.
c) Establish that, if $(a,b,c)\in\mathbf{C}^3$, $\widehat{abc}$ is a compact convex subset of $\mathbf{C}$ equipped with its usual topology.
d) With the same notation prove the existence of:
$$\delta(\widehat{abc}) = \max\left\{|z'-z| / (z,z')\in\widehat{abc}^2\right\}$$