# Minimum volume of intersection between two high-dim $\ell^1$-balls

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 $$B_1 \cap B_2$$?

• Distance in what norm? – fedja May 30 '19 at 1:46
• 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.) – Martin Sleziak May 30 '19 at 5:23
• What is the meaning of "minimum volume" here? Minimum with respect to different radius balls? – Sahil Kumar May 30 '19 at 7:11
• $L_1$, $L_2$, or $L_{infinity}$ – Alanwang May 30 '19 at 14:26
• 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. – Alanwang May 30 '19 at 17:31