Optimal muffin problem solutions occur on an infinite set of linear bounds. For each linear bound there is a continuous range of m/p where the bound is tight, plus a set of discrete optimal instances which converge to one end of the continuous optimal range. The muffin problem should really be considered in projective space r = m/p, and I'll henceforth usually write S(r) instead of S(m,p). Note that S(r) is often continuous in r, while neither of the alternate measures T or U are. We have T(a,b) = S(a,b)/min(a,b) and U(a,b) = abT(a,b). Some choices for unit measures which are continuous in r = a/b are the min, max, or some mean. Because the muffin problem is about the smallest size piece, min(a,b), used by S(a,b) and S(r), seems the appropriate choice of unit measure. Thus "muffins" (smaller parts) have size 1 and "persons" (larger parts) have size r. I noted earlier that S(m,p) <= m/4p for 3p <= m <= 4p. In projective space this result is S(r) <= r/4 for 3/2 <= r <= 2. The bound is tight on (3/2,8/5]. The discrete optimal solutions S(r) = r/4 occur at r = (12+8k)/(7+5k), e.g., S(12/7) = 3/7, S(5/4) = 5/12, S(28/17) = 7/17, ... which converge to S(8/5) = 2/5. I've written a program to calculate S'(r), which is the muffin problem restricted to partitions where all muffins are split into exactly 2 pieces. S'(r) differs from the general solution S(r) only at integral r, where S' = 1/2 and S = 1, and on some subintervals of (1,3/2), where S = 1/3 and S' < 1/3. Here is the beginning of a Fibonacci-like sequence of bounds from my program: S'(r) = (1r-0)/4 = (3/8,2/5] on (3/2,8/5] and at r = (4+8k)/(2+5k) (1r-0)/4 = (1-0r)/2 at r = 2/1 = 2.0000000 S'(2/1) = 1/2 = 0.50000000 S'(r) = (3-1r)/3 = [2/5,1/3) on [9/5,2/1) and at r = (12+9k)/(7+5k) (3-1r)/3 = (1r-0)/4 at r = 12/7 = 1.7142857 S'(12/7) = 3/7 = 0.42857143 S'(r) = (2r-1)/6 = (17/42,7/17] on (12/7,59/34] and at r = (21+59k)/(12+34k) (2r-1)/6 = (3-1r)/3 at r = 7/4 = 1.7500000 S'(7/4) = 5/12 = 0.41666667 S'(r) = (12-5r)/8 = [7/17,13/32) on [148/85,7/4) and at r = (80+148k)/(46+85k) (12-5r)/8 = (2r-1)/6 at r = 40/23 = 1.7391304 S'(40/23) = 19/46 = 0.41304348 S'(r) = (7r-6)/15 = (142/345,265/643] on (40/23,1119/643] and at r = (228+1119k)/(131+643k) (7r-6)/15 = (12-5r)/8 at r = 228/131 = 1.7404580 S'(228/131) = 54/131 = 0.41221374 S'(r) = (73-32r)/42 = [1107/2686,2267/5502) on [9349/5372,228/131) and at r = (1347+9349k)/(774+5372k) (73-32r)/42 = (7r-6)/15 at r = 449/258 = 1.7403101 S'(449/258) = 319/774 = 0.41214470 S'(r) = (61r-60)/112 = (11909/28896,35408/85913] on (449/258,149516/85913] and at r = (10696+149516k)/(6146+85913k) (61r-60)/112 = (73-32r)/42 at r = 764/439 = 1.7403189 S'(764/439) = 2533/6146 = 0.41213798 S'(r) = (1101-487r)/615 = [1555262/3773645,111271/269985) on [1313469/754729,764/439) and at r = (160212+6567345k)/(92059+3773645k) (1101-487r)/615 = (61r-60)/112 at r = 160212/92059 = 1.7403187 S'(160212/92059) = 37941/92059 = 0.41213787 I hope these cases provide some interesting examples for the mixed integer optimization folks to chew on :) I'll give some example optimal partitions from this sequence (40/23, 228/131, 449/258) in a subsequent message.