Geodesic
Convex Sets, Cantor Sets and a Midpoint Property
Canadian mathematical bulletin, Tome 19 (1976) no. 4, pp. 467-471

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It is well known that every point of the closed unit interval I can be expressed as the midpoint of two points of the Cantor ternary set D. See [2, p. 549] and [3, p. 105]. Regarding J as a one dimensional compact convex set, it seems natural to try to generalize the above result to higher dimensional convex sets. We prove in section 3 that every convex polytope K in Euclidean space R d contains a topological copy C of D such that each point of K is expressible as a midpoint of two points of C. Also, we give necessary and sufficient conditions on a planar compact convex set for it to contain a copy of D with the midpoint property above. In the final section we prove a result on minimal midpoint sets. 
Reiter, Harold. Convex Sets, Cantor Sets and a Midpoint Property. Canadian mathematical bulletin, Tome 19 (1976) no. 4, pp. 467-471. doi: 10.4153/CMB-1976-070-9
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     title = {Convex {Sets,} {Cantor} {Sets} and a {Midpoint} {Property}},
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