Classify a region by its slope
Jelly,  18 17  16 bytes
I believe there is a lot of potential for this effort to be out-golfed
L‘HạŒỤṀ€IṠQṢ«FE$

A monadic link accepting a list of lists of numbers which returns a list of integers:
Concave -&gt;  [0, 0]
Flat    -&gt;  [-1, 0, 1]
Convex  -&gt;  [-1, 0]
Mixed   -&gt;  [-1, 0, 0]

Try it online! Or see the test-suite.
L‘H could be replaced with the less efficient but atomically shorter JÆm.
How?
L‘HạŒỤṀ€IṠQṢ«FE$ - Link: list of (equal length) lists of numbers
L                - length
 ‘               - increment
  H              - halve
                 -   = middle 1-based index (in both dimensions as the input is square)
    ŒỤ           - sort multi-dimensional indices by their corresponding values
                 -   = a list of pairs of 1-based indexes
   ạ             - absolute difference (vectorises)
                 -   = list of [verticalDistanceToMiddle, horizontalDistanceToMiddle] pairs
      Ṁ€         - maximum of €ach
                 -   each = N/2-k (i.e. 0 as middle ring and N/2 as outermost)
        I        - incremental deltas (e.g. [3,2,2,3,1]-&gt;[3-2,2-2,3-2,1-3]=[-1,0,1,-2])
         Ṡ       - sign (mapping -n:-1; 0:0; and +n:1)
          Q      - de-duplicate
           Ṣ     - sort
                 -   = concave:[0, 1]; convex:[-1, 0]; flatOrMixed:[-1, 0, 1]
               $ - last two links as a monad
             F   -   flatten
              E  -   all equal? (1 if flat otherwise 0)
            «    - minimum (vectorises)
                 -   = concave:[0, 0]; convex:[-1, 0]; mixed:[-1, 0, 0]; flat:[-1, 0, 1]