Convex Combinations of List Elements

Question

Let's say I have two lists $L_1=\{a_1,...,a_5\}$ and $L_2=\{b_1,...,b_10\}$. Is there an easy way to make new lists from convex combinations of these two lists' entries? Let me be more specific:

1 Suppose the two lists $L_1,L_2$ are of equal length $n$; how do I create a new list whose entries are linear combinations of the entries in $L_1,L_2$? I first defined $f(x)=wx$ and $g(x)=(1-w)x$ and then used Map[(# + Map[f, list1]) &, {Map[g, list2]}]. This works but it's not very neat, is there another way? (Let's call such a hypothetical neat function $u$.)

2 Now let's go back to the lists of unequal length. I wanna be able to apply $u$ to sublists of $L_1$ and $L_2$ of length at most five. One could of course delete entries from the lists in question and then apply $u$, but I'm thinking there's gotta be a better way, but how? That is to say, how do I make a function where I can specify to map the first two entries of each list to a 2-list (producing the list $\{wa_1+(1-w)b_1,wa_2+(1-w)b_2\}$), or to map the second and third entries of each list to a 2-list? (producing the list $\{wa_2+(1-w)b_2,wa_3+(1-w)b_3\}$). On this one I'm completely lost.

Thanks in advance :)

EDIT: With h[x_,y_]:=wx+(1-w)y, Apply[h,{list1,list2}] answers question 1.

Answer 1

Perhaps (for equal length lists)

foo[h_, l1_, l2_, parts_] := h@@@Transpose[{l1, l2}][[parts]]
(* or foo[h_, l1_, l2_, parts_] := Apply[h, Transpose[{l1, l2}][[parts]], 1] *)

{l1, l2} = Array[#, {5}] & /@ {a, b};
h[x_, y_] := w x + (1 - w) y

foo[h, l1, l2, {2, 3}]
(* {w a[2] + (1 - w) b[2], w a[3] + (1 - w) b[3]} *)

?

Answer 2

If the length of list1 and list2 are the same, I think you can form the convex hull of the pointset formed by transposing them - since every point in a convex hull is a convex combination of some of these points.

list1 = RandomVariate[NormalDistribution[], 50];
list2 = RandomVariate[NormalDistribution[], 50];
pts = Transpose[{list1, list2}];
reg = ConvexHullMesh[pts]

You may now test any point {x,y} for membership in this convex hull by doing

{x,y}\[Element]reg

The coordinates in the convex hull are those that are convex combinations of list1 and list2 simultaneously. I assume this is what you really want. If you wanted any convex combination in either list1 or list2, this would just be a boring rectangle of dimensions {{Min[list1],Max[list1]},{Min[list2],Max[list2]}}

You can now generate random numbers, form pairs of them {a,b} and test for membership in the region. Then randomly choose a or b and put that in a list, rinse repeat, you have a convex combination list.