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 $B_1 \cap B_2$?
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$\begingroup$ Distance in what norm? $\endgroup$– fedjaCommented May 30, 2019 at 1:46
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$\begingroup$ 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 SleziakCommented May 30, 2019 at 5:23
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$\begingroup$ What is the meaning of "minimum volume" here? Minimum with respect to different radius balls? $\endgroup$– Sahil KumarCommented May 30, 2019 at 7:11
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$\begingroup$ $L_1$, $L_2$, or $L_{infinity}$ $\endgroup$– AlanwangCommented May 30, 2019 at 14:26
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$\begingroup$ 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. $\endgroup$– AlanwangCommented May 30, 2019 at 17:31
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