Let C be a plane convex body. The relative distance (or C-distance) of points \(a, b\in C\) is defined by the ratio of the Euclidean distance of a and b to the half of the Euclidean distance of \(a_{1}, b_{1}\in C\), where \(a_{1}b_{1}\) is a longest chord of C parallel to the line-segment ab. Denote by \(\phi _{k}(C)\) the greatest possible number d such that the boundary of C contains k points in pairwise C-distance at least d and denote by \({\mathcal {C}}\) the family of plane convex bodies. Let \(\phi _{k}({\mathcal {C}})=\sup \{\phi _{k}(C)\mid C\in {\mathcal {C}}\}\). In this paper we prove \(\phi _{9}({\mathcal {C}})=\sqrt{3}-1\).

令C为平面凸体。的相对距离（或Ç -距离）的点\（A，B \用C \）通过的欧几里德距离的比来定义一个和b到的欧几里德距离的一半\（A_ {1}，B_ {1} \ in C \），其中\（a_ {1} b_ {1} \）是平行于线段ab的C的最长弦。表示由\（\披_ {K}（C）\）的最大可能的数目d，使得边界Ç包含ķ在成对点Ç -distance至少d并用\（{\ mathcal {C}} \\）表示平面凸体的族。令\（\ phi _ {k}（{\ mathcal {C}}）= \ sup \ {\ phi _ {k}（C）\ mid C \ in {\ mathcal {C}} \} \）中。在本文中，我们证明\（\ phi _ {9}（{\ mathcal {C}}）= \ sqrt {3} -1 \）。