The localized feature energy is semiconvex along generalized geodesics

Regard $(x,y)$ as random variables on the probability space $({\mathbb{R}}^{3d},\pi )$ and set $X(z,x,y)=x$, $Y(z,x,y)=y$ in $L^2(\pi ;{\mathbb{R}}^d)$. The Euclidean label dimension is $3d$ and the output dimension is $d$; the arbitrary-label lift and Taylor statement apply independently of these dimensions. These are admissible $Q_r$-valued lifts by [A1], and $X_t=(1-t)X+tY$ lifts ${\mu }_t$ by [F3]. Therefore the feature and potential integrals defining $F({\mu }_t)$ equal the corresponding lifted integrals.

The gradient-Lipschitz estimate [F1] implies $X\mapsto R(\int \mathrm{\Phi }(X)d\pi )+\frac{C_r}{2}\mathrm{\parallel }X{\mathrm{\parallel }}_{L^2(\pi )}^2$ is convex on the segment: its derivative is monotone along the segment, since the gradient difference has norm at most $C_r$ times the segment displacement. The one-dimensional convex secant inequality gives ${\displaystyle R\text{\hspace{0.17em}⁣}(\int \mathrm{\Phi }(X_t)d\pi )\le (1-t)R\text{\hspace{0.17em}⁣}(\int \mathrm{\Phi }(X)d\pi )+tR\text{\hspace{0.17em}⁣}(\int \mathrm{\Phi }(Y)d\pi )+\frac{C_r}{2}t(1-t)\mathrm{\parallel }X-Y{\mathrm{\parallel }}_{L^2(\pi )}^2\mathrm{.}}$ By [F2], $\int V(X_t)d\pi \le (1-t)\int V(X)d\pi +t\int V(Y)d\pi -\frac{{\lambda }_V}{2}t(1-t)\mathrm{\parallel }X-Y{\mathrm{\parallel }}_{L^2(\pi )}^2$. Add these inequalities and use $\mathrm{\parallel }X-Y{\mathrm{\parallel }}^2=\int \mathrm{\mid }x-y{\mathrm{\mid }}^{2}d\pi$ to obtain the displayed claim.