Extrema of P-Invariant Functions on the Brillouin Zone

At the Colloquium in honor of Léon Van Hove it was very moving to speak of him and to hear all what was said of this great physicist, who died one year before, by his friend and his wife. This is an expanded version of the given lecture. It starts from the exposition of a paper that Léon Van Hove wrote nearly forty years ago. This paper was extremely original and important; it is still often quoted.