\\begin{prop}
\nLet $K$ be a compact convex set in $\\mathbb{R}^n$. For any $P\\in \\partial K$, there exists a support hyperplane of $K$ containing $P$.
\n\\end{prop}
\n\\begin{proof}For every $P\\in \\partial K$, take $f:~S^{n-1}\\rightarrow \\mathbb{R}$,
\n\\[
\nf(u)=h(K,u)-P\\cdot u, \\quad u\\in S^{n-1},
\n\\]
\nwhere $h(K,\\cdot)$ is support function of K. Thus, $f(u)\\geqslant0$ for any $u\\in S^{n-1}$. Since $h(K,\\cdot)$ is continuous on $S^{n-1}$ which is a compact subset of $\\mathbb{R}^{n-1}$, $f(\\cdot)$ is continuous and there exists $u_1\\in S^{n-1}$ such that
\n
\n\\[
\nf(u_1)=\\min_{u\\in S^{n-1}}f(u).
\n\\]
\nLet $\\rho=f(u_1)$. Thus, $\\rho\\geqslant0$.<\/p>\n
Suppose $\\rho>0$. Denote $B_P(\\frac{1}{2}\\rho)$ for the closed ball with center $P$ and radius $\\frac{1}{2}\\rho$, then, obviously, the support function of $B_P(\\frac{1}{2}\\rho)$ is
\n\\[
\nh(B_P(\\frac{1}{2}\\rho), u)=P\\cdot u+\\frac{1}{2}\\rho, \\quad u\\in S^{n-1}.
\n\\]
\nThus we have
\n\\[
\nh(k,u)-h(B_P(\\frac{1}{2}\\rho), u)=h(K,u)-P\\cdot u-\\frac{1}{2}\\rho\\geqslant\\frac{1}{2}\\rho>0, \\quad u\\in S^{n-1}.
\n\\]
\ni.e.,
\n\\[
\nh(k,u)\\geqslant h(B_P(\\frac{1}{2}\\rho), u), \\quad\\text{for any }u\\in S_{n-1},
\n\\]
\nwhich implies
\n\\[
\nB_P(\\frac{1}{2}\\rho)\\subset K.
\n\\]
\nThis contradicts the condition $P\\in \\partial K$. Therefore, we must have $\\rho=0$,\u00a0i.e., \u00a0$f(u_1)=h(K,u_1)-P\\cdot u_1=0$, which means $P\\in H(K,u_1)$ \u00a0and the desired results obtained.
\n\\end{proof}<\/p>\n","protected":false},"excerpt":{"rendered":"
\\begin{prop} Let $K$ be a compact convex …<\/p>\n