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The SVG was generated by running the Lua program given below. It deliberately fails if the closest points would be overlapping in the image, because that would be confusing.

local WD, HT, NUM_POINTS = 256, 256, 16
local COLOR = { normal = '#000', closest = '#f00' }
local RADIUS = 5

math.randomseed(os.time())
local P = {}
for _ = 1, NUM_POINTS do
    P[#P+1] = { x = RADIUS + (WD - 2*RADIUS) * math.random(),
                y = RADIUS + (HT - 2*RADIUS) * math.random() }
end

-- Find closest pair, using naive algorithm.
local closest_a, closest_b
local min_dist
for i, p in ipairs(P) do
    for j, q in ipairs(P) do
        if i ~= j then
            local dist = math.sqrt((p.x - q.x)^2 + (p.y - q.y)^2)
            if not min_dist or dist < min_dist then
                min_dist = dist
                closest_a, closest_b = i, j
            end
        end
    end
end

if min_dist < RADIUS then
    error("points on top of each other, run me again")
end

io.write('<?xml version="1.0" encoding="UTF-8" standalone="no"?>\n',
         '<svg svg="http://www.w3.org/2000/svg" version="1.0"',
         ' width="', WD, '" height="', HT, '">\n')
for i, p in ipairs(P) do
    local color = COLOR.normal
    if i == closest_a or i == closest_b then color = COLOR.closest end
    io.write(' <circle cx="', RADIUS + p.x, '" cy="', RADIUS + p.y,
             '" r="', RADIUS, '" style="fill:', color, '"/>\n')
end
io.write('</svg>\n')