Two main results are proven in this paper. Given an open convex set \(\Omega\subset {\mathbb R}^m\) of finite Lebesgue measure with Dirichlet eigenvalues \(\lambda_j(\Omega)\), \(j\in {\mathbb N}\), the following problem is considered: \[ I_k(c)=\inf\{\lambda_k(\Omega):\Omega\subset {\mathbb R}^m \text{ open and convex}, \mathcal{T}(\Omega)=c\}. \tag{*} \] Here \(\mathcal{T}\) is a function defined on the space of open convex sets in \({\mathbb R}^m\), satisfying some of the following hypothesis

(a)

\(\mathcal{T}\) is invariant under isometries, monotone (\(\Omega_1\subset\Omega_2\Rightarrow\mathcal{T}(\Omega_1)\leq \mathcal{T}(\Omega_2)\)), and non-negative (\(\mathcal{T}(\Omega)=0 \Leftrightarrow \Omega=\emptyset\)).

(b)

\(\mathcal{T}(\alpha\Omega)=\alpha^\tau\mathcal{T}(\Omega)\) for some \(\tau>0\) and every \(\alpha>0\).

(c1)

\(T^*=\inf\{\mathcal{T}(\Omega): \Omega\subset\mathbb{R}^m \text{ open and convex}, |\Omega|=1\}>0\).

(c2)

There exists an open convex set \(D\subset\mathbb{R}^m\) with \(|D|=1\), unique up to isometries, such that \(T^*=\mathcal{T}(D)\).

(d)

There exist constants \(K<\infty\) and \(t>1/\tau\) such that, if \(\Omega\subset\mathbb{R}^m\) is an open bounded convex set, then \(\text{diam}(\Omega)\leq K\mathcal{T}(\Omega)^t|\Omega|^{(1-t\tau)/m}\).

Under these assumptions, the author proves in Theorem 1: (i) if \(\mathcal{T}\) satisfies (a), (b) and (c1), then the variational problem (*) has a minimizer; (ii) if \(\mathcal{T}\) satisfies (a), (b), (c2) and (d), and \((\Omega^*_k)_{k\in\mathbb{N}}\) are minimizers of (*) for \(k\in\mathbb{N}\), then there exists a sequence of isometric copies of these minimizers \((\Omega^{**}_k)_{k\in\mathbb{N}}\) such that \[ \Omega^{**}_k\to \left(c\over{\mathcal{T}(D)}\right)^{1/\tau}D, \] in both the Hausdorff metric and the complementary Hausdorff metric. Taking \(\mathcal{T}\) as the Lebesgue measure, (a), (b) and (c1) are satisfied, but neither (c2) nor (d). The perimeter functional satisfies all properties (a), (b), (c1), (c2) and (d) (see the proof of Corollary 5) as well as the moment of inertia of an open set with respect to its center of mass (see the proof of Corollary 6).
The second result in this paper is Theorem 2, providing relations between \(\pi_k\), \(\mu_k\) and \(\omega_m\), for all integers \(m\geq 2\) and \(k\geq 1\). Here \(\omega_m\) is the Lebesgue measure of the unit ball in \(\mathbb{R}^m\), and \(\pi_k\) and \(\mu_k\) are defined by \[ \pi_k=\inf\{|\Omega|:\Omega\in\mathfrak{B}_k\},\quad\mu_k=\inf\{|\Omega|:\Omega\in \mathfrak{M}_k\}, \] where \(\mathfrak{B}_k\) is the set of minimizers of \(\inf\{\lambda_k(\Omega):\Omega\subset\mathbb{R}^m \text{ open}, \mathcal{P}(\Omega)=1,|\Omega|<\infty\}\), and \(\mathfrak{M}_k\) is the set of minimizers of \(\inf\{\lambda_k(\Omega):\Omega\subset\mathbb{R}^m \text{ quasi-open},|\Omega|=1\}\).