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I think this may help: Lemma. For any two convex sets, and any vector $x$, the function $F(t)={\mathrm{Vol}}\left(\mathrm{conv}\left(K\cup(L+xt)\right)\right)$ is convex, as a function of real variab …

This has nothing to do with circles (or polygons). If one convex curve is inside another, then the length of the inner curve is smaller.

The necessary and sufficient conditions are: $(1/n)\sum x_j=(1/3)(a+b+c)$, and all $x_j$ lie inside the closed triangle $(a,b,c)$. Proof of sufficiency. For every affine function, we have equality. L …

Your broken line closes iff the triangles $(v_1,v_2,v_3)$ and $(v_3,v_4,v_5)$ are equal, and $[v_1,v_3]$ is their common side. So there are two conditions: one is that $m_1=m_3$, by the hyperbolic law …

All that is needed is an estimate of the length of the piece of $\gamma$ in the disk $|z|\leq r$. This can be done as follows. Remove a bounded arc of $\gamma$ so that two unbounded pieces remain, an …

This paper is contained in the 1-st volume of Aleksandrov's Selected works. This has an English translation: Alexandrov, A. D. Reshetnyak, Yu. G. (ed.) Selected works. Part 1: Selected scientific pape …

If you have an arbitrary system of linear equations and add to it one equation, the dimension of the space of solutions can decrease by at most 1. Do take any $d$ equations of your hyperplanes, order …

Denote the diameter by $d$ and distance by $|x-y|$. Then there are $y,z$ such that $d=|y-z|$ and we have by triangle inequality for every $x$: $$d=|y-z|\leq |y-x|+|x-z|\leq 2f(x),$$ so we obtain your …