Basically, I can't solve Exercise 1.1 of chapter 2 of "Mathematics++, Selected Topics Beyond the Basic Courses" which is the following:
For a set $V= \{ v_1 , ..., v_N \}$, show that $conv(V) \subset \cup _{i\leq n} B(\frac{1}{2} v_i , \Vert \frac{1}{2} v_i \Vert)$.
The author attributes the result to Elekes, and I found a similar result in Elekes paper "A Geometric Inequality and the Complexity of Computing Volume" to by found on https://www.math.cmu.edu/~af1p/Teaching/MCC17/Papers/elevol.pdf (the claim in theorem 1).
I can't follow Elekes' proof though. He doesn't mention where the $S_i$ are centered. If they are centered at the center of $S$, his claim is false for $P_i$ equidistant from that center, as all the balls are the same and don't contain the $P_i$...
Can someone please help me solve the first exercise and get me out of confusion on the Elekes paper ? Thanks.