We have two responses for you

Tom,

With his insistence on precise definitions it sounds as if your son will make a good mathematician! The bad news is that his teacher is probably correct -- your son has the right definition but the wrong context. (However, it could be that the question was ambiguous, in which case your son could be correct; we would have to see the question for a definitive response.)
In mathematics there are a half dozen different places where the word "vertex" is used, each with a different (but similar) definition. When you are talking about a POLYHEDRON (and not just any three dimensional object!), a vertex is a point where three OR MORE edges meet. When you are talking about a cone, a vertex is the point where the straight lines that form the side of the cone meet.
Our web site has had many complaints about inconsistent definitions, about vertices in particular. For an earlier, more complete answer, see
mathcentral.uregina.ca/QQ/database/QQ.09.08/h/deb1.html

Chris

That isn't a standard definition. There are several definitions that are standard, each valid in its own context.

A vertex of a polyhedron (or equivalent shape in other dimensions) is a point where edges meet. In two dimensions this would always be two edges; in higher dimensions three or more (think of a pyramid).

A vertex of a curve is a point where the curvature is higher than anywhere else nearby (the ends of an ellipse, for instance.)

For a general convex body, a vertex is often defined to be a point at which the intersection of all the supporting hyperplanes there is the point. A hyperplane is a line in the plane, a plane in 3D space, etc. It is "supporting" if it contains the point but no point interior to the body.

My ten year old son argues that the point at the top of a cone is not a vertex
since it does not fit the definition. He got the answer wrong on a test recently
but insists that he is right. I need a mathematician to answer this for him.

For a cone, the last definition above would be appropriate, and your son's answer would be wrong. However:

1. he is clearly thinking about the problem; even if he does not get the points he gets a "good try" from me.
   
   
2. the class were probably not given the definition I gave you; he may (or may not) be correct under the definition that the class were given. If the class were given no definition (all too frequent in the modern classroom) and expected to "induce" one from examples, was a cone among the examples?

Good Hunting!
RD