Let $B_1$ and $B_2$ be two balls with the same radius, in $\mathbb R^n$ with the $\ell^1$ norm. The distance between the centers of $B_1$ and $B_2$ is $d(B_1, B_2)$. Is there any deterministic method or probabilistic method to calculate the minimum volume of $B_1 \cap B_2$?

  • $\begingroup$ Distance in what norm? $\endgroup$ – fedja May 30 '19 at 1:46
  • $\begingroup$ I will point out that the (geometry) tag is deprecated - see the tag-info. (However, I'll leave the choice of a suitable geometry-related top-level tags to more experienced users.) $\endgroup$ – Martin Sleziak May 30 '19 at 5:23
  • $\begingroup$ What is the meaning of "minimum volume" here? Minimum with respect to different radius balls? $\endgroup$ – Sahil Kumar May 30 '19 at 7:11
  • $\begingroup$ $L_1$, $L_2$, or $L_{infinity}$ $\endgroup$ – Alanwang May 30 '19 at 14:26
  • $\begingroup$ The distance can be $L_1$ or $L_2$. The two balls have the same radius. Given the fixed distance, the two balls can have different overlaps. The aim is to find the minimum overlap between the two balls. $\endgroup$ – Alanwang May 30 '19 at 17:31

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