Functionals of probability measures are rarely convex under ordinary mixture. The geometry relevant to transport follows displacement geodesics, and convexity along those curves controls uniqueness, contraction, inequalities, and evolution equations.
Geodesic \(\lambda\)-convexity
Definition. A functional \(\mathcal F:\mathcal P_2(\mathbb R^d)\to(-\infty,\infty]\) is displacement \(\lambda\)-convex if for every pair \(\mu_0,\mu_1\) in its effective domain there exists a constant-speed geodesic \(\mu_t\) such that \[ \mathcal F(\mu_t) \le(1-t)\mathcal F(\mu_0)+t\mathcal F(\mu_1) -\frac{\lambda}{2}t(1-t)W_2^2(\mu_0,\mu_1). \] When the inequality holds along every geodesic, one says strongly displacement convex.
Positive \(\lambda\) is strong convexity, \(\lambda=0\) is ordinary geodesic convexity, and negative \(\lambda\) allows controlled concavity. The correction term is the metric analogue of strong convexity in a Hilbert space.
Potential energy
Proposition. Let \[ \mathcal V(\mu)=\int_{\mathbb R^d}V(x)\,\mathrm{d}\mu(x). \] If \(D^2V\ge\lambda I\), then \(\mathcal V\) is displacement \(\lambda\)-convex.
Proof. Let \(\pi\) be an optimal coupling and \(\mu_t=((1-t)x+ty)_\#\pi\). Pointwise \(\lambda\)-convexity of \(V\) gives \[ V((1-t)x+ty) \le(1-t)V(x)+tV(y)-\frac{\lambda}{2}t(1-t)|x-y|^2. \] Integrate with respect to \(\pi\), and use \(\int|x-y|^2\,\mathrm{d}\pi=W_2^2(\mu_0,\mu_1)\).
Interaction energy
Proposition. Let \(W:\mathbb R^d\to\mathbb R\) be even and convex, and define \[ \mathcal W(\mu)=\frac12\iint W(x-y)\,\mathrm{d}\mu(x)\,\mathrm{d}\mu(y). \] Then \(\mathcal W\) is displacement convex. If \(D^2W\ge\lambda I\), the corresponding convexity estimate holds modulo translations; the center-of-mass direction remains flat.
Proof. Use two independent copies \((x,y)\), \((x',y')\) of an optimal coupling. Along interpolation, \[ x_t-x_t'=(1-t)(x-x')+t(y-y'). \] Convexity of \(W\), followed by integration over the product coupling, gives \[ \mathcal W(\mu_t)\le(1-t)\mathcal W(\mu_0)+t\mathcal W(\mu_1). \] Strong convexity yields an additional quadratic term involving \(|(x-x')-(y-y')|^2\); it vanishes for common translations, explaining the flat direction.
Internal energy and McCann's condition
Definition. For \(\mu=\rho\,\mathrm{d}x\), define \[ \mathcal U(\mu)=\int_{\mathbb R^d}U(\rho(x))\,\mathrm{d}x, \] and set \(\mathcal U(\mu)=+\infty\) when \(\mu\) is not absolutely continuous. McCann's condition is that \[ s\longmapsto s^dU(s^{-d}) \] is convex and nonincreasing on \((0,\infty)\), together with the usual lower-semicontinuity growth assumptions.
McCann displacement-convexity theorem. If \(U\) satisfies McCann's condition, then \(\mathcal U\) is displacement convex on \(\mathcal P_2(\mathbb R^d)\).
Proof for smooth positive densities. Let \(T=\nabla u\) be the optimal map from \(\rho_0\) to \(\rho_1\), and \(T_t=(1-t)I+tDT\). The Jacobian equation gives \[ \rho_t(T_t(x))=\frac{\rho_0(x)} {\det((1-t)I+tDT(x))}. \] Since \(DT\) is positive semidefinite, concavity of \(\det(\cdot)^{1/d}\) on positive matrices yields \[ \det((1-t)I+tDT)^{1/d} \ge(1-t)+t\det(DT)^{1/d}. \] After changing variables \(z=T_t(x)\), the internal energy becomes an integral of \[ J\,U(\rho_0/J),\qquad J=\det((1-t)I+tDT). \] McCann's condition is exactly the scalar convexity and monotonicity needed to combine the determinant inequality with the endpoint Jacobians. Integration proves the geodesic convexity inequality. Approximation by smooth positive densities and lower semicontinuity give the general case.
Entropy and porous-medium energies
Boltzmann entropy. For \(U(r)=r\log r\), \[ s^dU(s^{-d})=-d\log s, \] which is convex and decreasing. Hence \[ \operatorname{Ent}(\mu)=\int\rho\log\rho\,\mathrm{d}x \] is displacement convex.
Power energies. For \(U(r)=r^m/(m-1)\), McCann's condition holds when \[ m\ge1-\frac1d. \] The range includes the porous-medium case \(m>1\) and part of the fast-diffusion range.
Free energies
The functional \[ \mathcal F(\mu) =\int U(\rho)\,\mathrm{d}x +\int V\,\mathrm{d}\mu +\frac12\iint W(x-y)\,\mathrm{d}\mu(x)\,\mathrm{d}\mu(y) \] combines internal, confining, and interaction energies. The previous results allow their displacement-convexity constants to be added. This single class generates the heat, Fokker--Planck, porous-medium, and aggregation--diffusion equations studied on pages 17 and 18.
Consequences of strict convexity
Uniqueness of minimizers. If \(\mathcal F\) is displacement \(\lambda\)-convex with \(\lambda>0\), then it has at most one minimizer. If a minimizer \(\mu_\star\) exists, then \[ \mathcal F(\mu)-\mathcal F(\mu_\star) \ge\frac{\lambda}{2}W_2^2(\mu,\mu_\star). \]
Proof. Apply \(\lambda\)-convexity along a geodesic from \(\mu_\star\) to \(\mu\). Minimality gives \(\mathcal F(\mu_\star)\le\mathcal F(\mu_t)\). Rearranging, \[ t\{\mathcal F(\mu)-\mathcal F(\mu_\star)\} \ge\frac{\lambda}{2}t(1-t)W_2^2(\mu,\mu_\star). \] Divide by \(t\) and let \(t\downarrow0\). Two minimizers would make the left side zero and hence have zero distance.
Generalized geodesics
The squared Wasserstein distance is not convex along every ordinary geodesic in dimensions \(d\ge2\). Many variational arguments use generalized geodesics with a common base measure: glue optimal maps from a base \(\omega\) to \(\mu_0,\mu_1\), then interpolate their images. Along these curves, squared distance to \(\omega\) has the exact Hilbertian quadratic identity needed for the JKO scheme.