<p>The inner radius of the outer child may be equal to zero due to the following code</p>

<pre><code>  VantagePointTree* tree = this;

  while (tree-&gt;Parent() != NULL)
  {
    tree-&gt;Parent()-&gt;Bound() |= tree-&gt;Bound();
    tree-&gt;Parent()-&gt;furthestDescendantDistance = 0.5 *
            tree-&gt;Parent()-&gt;Bound().Diameter();
    tree = tree-&gt;Parent();
  }
</code></pre>

<p>(we have to ensure that each parent contains its children), there is no errors.</p>

<p>It seems I do not use <code>FirstSiblingFirstPointIsCentroid</code> correctly. I have to think about it.</p>

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