Given $n$ balls in $\mathbb{R}^d$ with radii $r_1,r_2,\dots,r_n$. Assume that this system of balls can not be separated by a hyperplane (that is, if a hyperplane $H$ does not intersect these balls, they necessary belong to the same half-space bounded by $H$). Prove that all $n$ balls may be covered by a ball of radius $\sum r_i$.

reference: A. W. Goodman and R. E. Goodman, A circle covering theorem, Amer. Math. Monthly 52 (1945), 494-498.

Given $n$ balls in $\mathbb{R}^d$ with radii $r_1,r_2,\dots,r_n$. Assume that this system of balls can not be separated by a hyperplane (that is, if a hyperplane $H$ does not intersect these balls, they necessarily belong to the same half-space bounded by $H$). Prove that all $n$ balls may be covered by a ball of radius $\sum r_i$.

reference: A. W. Goodman and R. E. Goodman, A circle covering theorem, Amer. Math. Monthly 52 (1945), 494-498.

Given $n$ balls in $\mathbb{R}^d$ with radii $r_1,r_2,\dots,r_n$. Assume that this system of balls can not be separated by a hyperplane. Prove that it may be covered by a ball of radius $\sum r_i$.

reference: A. W. Goodman and R. E. Goodman, A circle covering theorem, Amer. Math. Monthly 52 (1945), 494-498.

Given $n$ balls in $\mathbb{R}^d$ with radii $r_1,r_2,\dots,r_n$. Assume that this system of balls can not be separated by a hyperplane (that is, if a hyperplane $H$ does not intersect these balls, they necessary belong to the same half-space bounded by $H$). Prove that all $n$ balls may be covered by a ball of radius $\sum r_i$.

reference: A. W. Goodman and R. E. Goodman, A circle covering theorem, Amer. Math. Monthly 52 (1945), 494-498.