Discrete notions of curvatures are by now quite far developed, partially motivated by applications in shape-space analysis, computer vision, and physical simulation. When establishing discrete notions of curvatures, one is often guided by mimicking structural properties of the classical smooth setting and by the desire to recover classical notions in the limit of refinement.
While we have gained quite a bit of knowledge about consistent discrete notions of curvatures and differential operators by now, one question remains widely open and poses interesting challenges: Do the resulting minimizing shapes actually approximate their smooth counterparts or do they just provide us with pretty pictures?
Building on tools from variational analysis, I will discuss some aspects of convergence of discrete minimizers of geometric energy functionals. As a particular example, I will discuss convergence of two of the perhaps most prominent minimizers of geometric energy functionals: discrete minimal surfaces and discrete Euler elasticae.
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