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Expected Number of Points on a Sphere such that their Convex Hull Contains the Origin

In [1]:
def random_unit_vector(dimension, distribution): v = vector(distribution.get_random_element()) v = [QQ(x) for x in v] return v @parallel def count_points_required(dimension, distribution): points = [] while True: points.append(random_unit_vector(dimension, distribution)) if len(points) <= dimension: continue convex_hull = Polyhedron(vertices=points) if vector(QQ, dimension) in convex_hull: return len(points)
In [2]:
import IPython.display def plot_mark(x, color, legend): return arrow((x, -1e-10), (x, 0), color=color, legend_label=legend, legend_color=color) def plot_histogram(counts, dimension): average = mean(counts) expected = 2*dimension + 1 title = ("For d={} found average {}, expected {} after {} trials" .format(dimension, N(average, digits=5), N(expected, digits=5), len(counts))) bins = srange(min(counts) - 1/2, max(counts) + 1) G = histogram(counts, bins=bins, title=title) G += plot_mark(average, "red", "average") G += plot_mark(expected, "green", "expected") return G def test_conjecture(dimension, trials=10): distributions = [create_distribution() for x in range(trials)] test_cases = zip([dimension]*trials, distributions) counts = count_points_required(test_cases) counts = [output for (input, output) in counts] plot_histogram(counts, dimension).show() def test_conjecture_interactive(dimension, trials=10, distribution=None): counts = [] while True: test_cases = [(dimension, SphericalDistribution(dimension)) for x in range(trials)] counts_ = count_points_required(test_cases) counts += [output for (input, output) in counts_] H = plot_histogram(counts, dimension) IPython.display.clear_output(wait=True)
In [3]:
test_conjecture_interactive(3, trials=32, distribution='sphere')
WARNING: Some output was deleted.


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