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Suppose $P_1,P_2,,\cdots,P_n$ aren points inn the plane with given centriod $C=(a,b)$and $$d=\text{max distance}(P_i,P_j),i \neq j$$.Can we find a circle of some radius and center in terms of $d$ and $a$ and $b$ which contains the convex hull of the given points?
enter image description here Any hints/suggestions are highly appreciated.

AgnostMystic
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  • Doesn't the circle of radius $d$ about $C$ work? – Greg Martin Jun 17 '22 at 07:27
  • Oh ,I am so stupid!Thank you for pointing it out.By the way,can we improve it ,I mean can we make it any smaller? – AgnostMystic Jun 17 '22 at 07:47
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    I don't think so. Imagine you have many points very close together and one point "far away." With this setting $d$ will be the distance between this stray point and one of the others. On the other hand, the centroid will be very close to the mass of points, so you'll need a radius very close to $d$ in order to include the outlier. – Patricio Jun 17 '22 at 09:18
  • A second thought revealed to me that it is not obvious.C may not necessarily be one of the points ,and hence any point being within a distance of $d$ is not obvious – AgnostMystic Jun 17 '22 at 09:44
  • Suppose for the sake of contradiction that some point $Z$ is outside the circle of radius $d$ about $C$. By definition of $d$, all other points lie within the circle of radius $d$ about $Z$. Is it then possible for the centroid to be $C$? – Greg Martin Jun 17 '22 at 18:18
  • I am not able to figure out.Seems not possible but could you kindly explain – AgnostMystic Jul 08 '22 at 07:12

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