Convexity of Sub-polygons of Convex Polygons
  A convex polygon is defined as a sequence (V_0,...,V_{n-1}) of points on a plane such that the union of the edges [V_0,V_1],..., [V_{n-2},V_{n-1}], [V_{n-1},V_0] coincides with the boundary of the convex hull of the set of vertices {V_0,...,V_{n-1}}. It is proved that all sub-polygons of any convex polygon with distinct vertices are convex. It is also proved that, if all sub-(n-1)-gons of an n-gon with n\ge5 are convex, then the n-gon is convex. Other related results are given. 
Iosif Pinelis