On logarithmic smoothing of the maximum function

We consider the maximum function f resulting from a finite number of smooth functions. The logarithmic barrier function of the epigraph of f gives ris e to a smooth approximation g(epsilon) of f itself, where epsilon > 0 denot es the approximation parameter. The one-parametric family g(epsilon) conver ges - relative to a compact subset - uniformly to the function f as epsilon tends to zero. Under nondegeneracy assumptions we show that the stationary points of g(epsilon) and f correspond to each other, and that their respec tive Morse indices coincide. The latter correspondence is obtained by estab lishing smooth curves x(epsilon) of stationary points for g(epsilon), where each x(epsilon) converges to the corresponding stationary point of f as ep silon tends to zero. In case of a strongly unique local minimizer, we show that the nondegeneracy assumption may be relaxed in order to obtain a smoot h curve x(epsilon).