Return dynamics on class O_C domains with fixed convex core converge globally like adaptive gradient descent of the thickness function, with fixed points equal to thickness critical points.
The Hornich-Hlawka functional inequality for functions with positive differences
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abstract
We analyze the role played by $n$-convexity for the fulfillment of a series of linear functional inequalities that extend the Hornich-Hlawka functional inequality, $f\left( x\right) +f\left( y\right) +f\left( z\right) +f\left( x+y+z\right) \geq f\left( x+y\right) +f\left( y+z\right)+f\left( z+x\right) +f(0),$ including extensions to the case of positive operators.
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Global Convergence of the Return Dynamics in the Class $\mathcal{O}_C$
Return dynamics on class O_C domains with fixed convex core converge globally like adaptive gradient descent of the thickness function, with fixed points equal to thickness critical points.