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3

Let's call the center of one fixed circle $O_1$ and the other $O_2$. Say the radius of the variable circle is $r$. Because the variable circle maintains tangency with the fixed circles we have the following equations: $$\overline{CO_1}=a+r$$ $$\overline{CO_2}=b-r$$ And adding the two we have:$$\overline{CO_1}+\overline{CO_2}=a+b$$ Hence we conclude that ...

3

without more indication about constraints, a basic solution would be to precompute a decomposition of the convex hull into a tiling of simplexs, then to compute the distance to the convex hull that is the min of distances from $y$ to each simplex. If $y$ can't be inside the convex hull, simplications are possible, like, consider only the distance to the ...

1

Taking the $\max$-norm, for example, the calculation of the distance is LP: $$\min\epsilon\qquad$$ \begin{align} &-\epsilon \mathbb{1}\le\sum_{i=1}^n\lambda_i x_i-y\le\epsilon\mathbb{1},\\ &\sum_{i=1}^n\lambda_i=1,\quad\lambda_i\ge 0,\quad i=1,\ldots,n. \end{align} where $\mathbb{1}$ is the vector of all ones. The unknowns are $\epsilon$ and all ...

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