As I know the answer to this problem is "no"...

Let $n$ be a large even number, and take the union of the vertices of two regular polygons with $n/2$ vertices, which are homothetic from their common center by a factor $r=1+\epsilon$ for some tiny $\epsilon$ (take $n/2$ to be odd if you want no three points to be collinear). This set naturally splits into pairs of points corresponding under the homothety. If $\epsilon$ was chosen small enough, any quadrilateral not containing both points from such a pair will be convex. The number of quadrilaterals that do contain a pair is $O(n^3)$, so the $99\%$ condition indeed holds if $n$ is large enough. Now, a convex polygon with more than $90\%$ of the vertices must contain both points of some pair. Consider the point closer to the center, and draw a line through it so that the convex polygon is contained in one side of the line. This already rules out at least $n/2+O(1)$ that are on the other side of the line that cannot be part of the polygon. $\blacksquare$

Phorphyrion wrote: Let $n$ be a large even number, and take the union of the vertices of two regular polygons with $n/2$ vertices, which are homothetic from their common center by a factor $r=1+\epsilon$ for some tiny $\epsilon$ (take $n/2$ to be odd if you want no three points to be collinear). This set naturally splits into pairs of points corresponding under the homothety. If $\epsilon$ was chosen small enough, any quadrilateral not containing both points from such a pair will be convex. The number of quadrilaterals that do contain a pair is $O(n^3)$, so the $99\%$ condition indeed holds if $n$ is large enough. Now, a convex polygon with more than $90\%$ of the vertices must contain both points of some pair. Consider the point closer to the center, and draw a line through it so that the convex polygon is contained in one side of the line. This already rules out at least $n/2+O(1)$ that are on the other side of the line that cannot be part of the polygon. $\blacksquare$ Sir How did you learn to solve this kind of geometry?Which book? I didn't find this type of geo in any book. How can I learn this type of geo?