Here's one way. You can also use pdist2() if you have it.
inpict = imread('https://www.mathworks.com/matlabcentral/answers/uploaded_files/1227937/image.png');
% binarize it somehow
mask = rgb2gray(inpict)>200;
% get only the interior boundary curves
mask = padarray(mask,[1 1],1,'both'); % pad
mask = bwperim(mask); % get perimeter
mask = mask(2:end-1,2:end-1); % crop
% get x,y coordinates of all pixels in the curves
CC = bwconncomp(mask);
[y1 x1] = ind2sub(size(mask),CC.PixelIdxList{1});
[y2 x2] = ind2sub(size(mask),CC.PixelIdxList{2});
% find distance between all points
D = sqrt((x1-x2.').^2 +(y1-y2.').^2);
[minD idx] = min(D,[],'all','linear')
% find corresponding points on each curve
[r c] = ind2sub(size(D),idx);
pointlist = [x1(r) y1(r); x2(c) y2(c)];
% show the shortest distance
imshow(mask); hold on
plot(pointlist(:,1),pointlist(:,2))
Note that this finds the minimum euclidean distance rather than simply finding the minimum horizontal width of the dark region. The latter would be simpler, but I'm assuming that's not what you want.
