Abstract. In this talk I present some recent results about the minimization of $\lambda_k$ under diameter constraint. After providing existence, attained at a constant width body, and optimality conditions in any dimension, I focus my attention on the optimality of the disk in the plane, giving the precise list of 17 eigenvalues for which the disk is a local minimum. This last fact is confirmed by numerical simulations, which show non circular minimizers out of the afore mentioned 17 values of $k$. These results are obtained in collaboration with B. Bogosel (CMAP) and A. Henrot (IECL).
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