Answer:
Here's a Harvard-level math problem:
Suppose we have a set of N points in d-dimensional space, with each point represented as a vector x_i = (x_{i1}, x_{i2}, ..., x_{id}). Let d >= 2 be an integer. Define the "average distance" between two points in this set to be the quantity:
D = (1/N^2) * sum_{i=1}^N sum_{j=1}^N ||x_i - x_j||
where ||x_i - x_j|| denotes the Euclidean distance between x_i and x_j.
Prove that for any set of N points in d-dimensional space, we have:
D >= ||x_bar - x||
where x_bar = (1/N) * sum_{i=1}^N x_i is the centroid of the set, and x = (x_{1}, x_{2}, ..., x_{d}) is an arbitrary point in the space.
This problem involves concepts from linear algebra, calculus, and geometry, and requires a deep understanding of the properties of Euclidean distance, centroids, and vector spaces. A possible approach to solving it would be to use the triangle inequality to establish lower bounds on the average distance, and then use properties of centroids and vector spaces to derive the desired inequality. Overall, this problem requires a high level of mathematical sophistication and creativity, and would be suitable for advanced undergraduate or graduate students in mathematics.