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RESULT #1

S =

    10     3     6     1    11    20    25     3
     2     1     5     6     7     4     8    11

Unfortunately a convex pentagon can be formed by connedcting CCW orriented lines using the points (5.000000),
(3.000000),(2.000000),(1.000000),(4.000000). 

RESULT #2

S =

    22    10    18    -7    -3    20     2     9
    -1     5    27    11    26     7    27    13
    
Unfortunately a convex pentagon can be formed by connedcting CCW orriented lines using the points (1.000000),
(4.000000),(2.000000),(5.000000),(3.000000).

RESULT #3

S =

     0     0    10     4     4     6     6    10
     0    10     0     4     6     4     6    10
    
Unfortunately a convex pentagon can be formed by connedcting CCW orriented lines using the points (1.000000),
(4.000000),(2.000000),(5.000000),(3.000000). 

RESULT #4

S =

     0     0    10     4     4     5     5    10
     0    11     0     4     8     4     8    11

CONGRATULATIONS, your selected set of points has NO convex  5 -gons!!