A continuum temperature singularity for a radial pair potential

Proof. Suppose that five such points existed. Scale by e_t^{-1} and form the Gram matrix G of the four vectors from one point to the other four. Write \varepsilon=2\delta+\delta^2. The diagonal entries differ from 1 by at most \varepsilon. By the polarization identity, each off-diagonal entry differs from 1/2 by at most \varepsilon. The matrix G_0 with diagonal 1 and off-diagonal 1/2 has smallest eigenvalue 1/2, whereas

Thus G would have rank four, which is impossible for four vectors in \R^3.

For completeness, we give the weighted proof of the required graph bound, an instance of the Motzkin–Straus identity [8, Theorem 1]. For a graph with no complete subgraph on five vertices, maximize

Choose a maximizer with the smallest possible support. If two vertices in its support are nonadjacent, transferring all their combined weight to the one with the larger weighted neighbor sum cannot decrease the objective. It decreases the support, a contradiction. The support is therefore a clique with at most four vertices. At this maximizer the objective is

Applying this inequality to the weights a_i=1/n gives the asserted edge count. The case n=0 is immediate.□

Proof. The diameter of a cell is \sqrt3/4<1. Among the n_i(n_i-1)/2 pairs inside cell i, at most 3n_i^2/8 lie in the shell by Lemma 2.1. Every other pair has core energy at least E. Thus the core energy inside this cell is at least

All remaining core interactions are nonnegative. Also,

where the first bound includes harmless self-particle terms and the second follows from symmetry, (2.4), and 2n_i n_j\le n_i^2+n_j^2. This proves (2.5). The permitted deletions preserve every core interaction within a small cell. Reducing a negative term only increases the energy, so the same argument proves the last assertion.□

Proof. We give the details of the canonical limit first. Two finite-volume bounds control the density adjustments in its construction.

A trial bound on bounded density intervals. Fix D_0<\infty and choose a grid spacing q\in(0,1/2) with q^{-3}>D_0+1. For large L, at least D_0L^3 grid centers lie in \Lambda_L at distance q/8 from its boundary. For n\le D_0L^3, put one particle in each of n distinct balls of radius q/8 around such centers. Every pair has distance at least 3q/4, and each particle has at most Cq^{-3} neighbors at distance less than one. The core is bounded on [3q/4,1], so the positive energy of these configurations is at most E_q n for a finite constant E_q. We need not count the negative energy. The n! assignments of the labeled particles to the balls have disjoint integration regions and cancel the canonical factorial. If v_q>0 is the volume of a ball, this gives

Consequently

The constant is uniform over the physical class, and can be chosen uniformly for \beta in a compact subset of (0,\infty).

A lower bound. There are at most (L/b+2)^3 cropped cells in \Lambda_L. Cauchy–Schwarz in (2.5) and the inequality n!\ge(n/e)^n therefore give, with d_L=n/L^3,

Dropping the positive quadratic term also gives the same inequality with only its last two terms. In particular, Z_{L,n} is positive and finite, and a_L is bounded above and below when the density stays in a fixed bounded interval and L is large.

Convergence at a positive density. Fix \beta>0 and d>0. The increasing infima

converge to a finite number I. Select \ell_j\ge j and n_j in this set such that a_{\ell_j}(\beta,n_j)\le I_j+1/j. Then n_j/\ell_j^3\to d and a_{\ell_j}(\beta,n_j)\to I. Every sequence with L\to\infty and n/L^3\to d has liminf at least I_j for each j, and hence at least I.

To prove the reverse inequality, tile part of a large cube by

cubes of side \ell_j, leaving gaps of width two. Every interaction between distinct cubes is nonpositive. For any prescribed counts m_1,\ldots,m_k, restricting the integral to these cubes gives

Indeed there are (\sum_i m_i)!/\prod_i m_i! assignments of labels to the cubes, and these give exactly the factorials on the right.

Start with n_j particles in every small cube. Define a higher count n_{\mathrm{hi}}=\lfloor(4d+4)\ell_j^3\rfloor and the diluted densities

For large j, n_j>0, n_{\mathrm{hi}}>n_j, and d_b\to d, d_{\mathrm{hi}}\to4d+4. If a target count n exceeds kn_j, change some cubes to n_{\mathrm{hi}}; if it is smaller, empty some cubes. One additional intermediate count realizes n exactly. For fixed sufficiently large j, this is possible for all large L with n/L^3\to d. The limiting upper fraction of changed cubes is at most

All new counts are bounded by (4d+4)\ell_j^3, so their values of a_{\ell_j} have a common upper bound by (2.11). The baseline values are bounded in absolute value. Applying (2.13), taking the large-volume limit with j fixed, and then letting j\to\infty therefore gives

The single intermediate cube contributes zero in the first limit. This proves the all-sequence convergence for d>0.

Zero density and convexity. When n/L^3\to0, the linear lower bound derived above has limit zero. For an upper bound, fix a sufficiently large side \ell and use separated cubes as in (2.13), with empty baseline. Fill some cubes with \lfloor\ell^3\rfloor particles and at most one with an intermediate count. Their number is o(L^3), so (2.11) gives a limiting upper bound zero. Thus H_\varphi(\beta,0)=0.

For convexity, take 0\le d_1<d_2 and a density strictly between them. For a large fixed \ell, use counts \lfloor d_1\ell^3\rfloor and \lfloor d_2\ell^3\rfloor in the separated cubes. The desired density lies strictly between their diluted densities for all sufficiently large \ell. Choose their proportions to attain the target count, with at most one intermediate cube. Inequality (2.13), followed first by L\to\infty and then by \ell\to\infty, gives the convexity inequality for H_\varphi(\beta,\cdot). The endpoint cases of that inequality are immediate.

The function is finite and convex on [0,\infty), hence continuous on (0,\infty). The linear lower bound gives lower semicontinuity at zero; since its value there is zero, extension by +\infty to negative densities is a closed convex function. For each fixed density, Hölder’s inequality makes -\log Z_{L,n}(\beta) concave in \beta. Passage to the limit gives finite concavity of H_\varphi(\cdot,d) on (0,\infty). Finally (2.12) gives (2.8).

We have established the canonical limit with enough control over particle number to pass to the grand-canonical ensemble. In particular, the density variable in the next maximization will not require a separate approximation of the canonical free energy.

Pressure and conjugacy. On a compact subset of \Omega, (2.12) implies that, for sufficiently large D and all large L,

with c>0 independent of the potential in the class. The sum of these terms is exponentially small in L^3, whereas the term n=0 is one. Among the remaining at most \lfloor DL^3\rfloor+1 terms, choose one of maximal size. Every subsequence has a further subsequence on which its density converges to some d\in[0,D]. The all-sequence canonical limit identifies the limiting exponent as hd-H_\varphi(\beta,d). Conversely, the term n=\lfloor dL^3\rfloor supplies this exponent as a lower bound for every d\ge0. This proves (2.9).

The finite-volume logarithms are finite and jointly convex in (\beta,h), by Hölder’s inequality applied to their defining integrals and sums. The limit is finite by the density bound, and it is nonnegative because the empty configuration is allowed. Lemma A.1, the finite-dimensional convex convergence principle of [10, Corollary 7.18], upgrades pointwise convergence to uniform convergence on compact subsets of the open convex set \Omega. Finally, the closed convexity in density proved above permits the one-dimensional Fenchel biconjugacy identity, proved in Lemma A.2 and treated in [10, Theorem 11.1], yielding (2.10).□

Proof. The value at d=0 is zero. By (2.8), the objective in (2.14) is bounded above by

Condition (2.6) gives a uniform positive lower bound on \kappa/t-\sigma/2. On a compact set of (\beta,h) with \beta bounded away from zero, the right side is therefore negative for every sufficiently large d, uniformly in the potential and in \sigma. This proves the common density bound and the finite upper bound for the supremum. The objective is upper semicontinuous in d, since H_Q(\beta,\cdot) is lower semicontinuous. It attains its maximum on the resulting compact interval.

For each fixed d\ge0, the function in braces in (2.14) is convex in (\beta,h): the canonical function is concave in \beta, and the other terms are affine. Their supremum is convex. Its monotonicity in h follows from d\ge0, and its nonnegativity from the zero-density choice.□

Proof. Condition on the configuration Y outside C_i, and put

The interactions involving the n particles inside C_i are bounded below by

The empty cell contributes one to the conditional partition function. Its volume is at most b^3, also when cropped by the cube boundary. It follows that

Indeed, \beta nA_i\le\beta A_0n^2/4+\beta A_i^2/A_0; after this estimate the remaining negative quadratic coefficient is at least

The resulting one-variable series is bounded uniformly on \mathcal K. The conditional bound still depends on Y. We remove that dependence by a finite-volume bootstrap.

Let F_i=\mathbb E[e^{c n_i^2}]. These numbers are finite before the bootstrap: multiplying the grand canonical integrand by e^{cn_i^2} leaves a strictly negative quadratic in all cell counts, by (2.5). Cauchy–Schwarz with weights J_{ij} gives

If M>0, set \theta_{ij}=J_{ij}/M and \alpha=4(M/A_0)^2<1. Then c\alpha=\beta M^2/A_0. The row weights sum to at most one; place any missing weight on a value zero. Convexity of the exponential and then concavity of x^\alpha give

There are finitely many cells, so their maximum is bounded by a constant depending only on C_1,\alpha. When M=0, the conditional estimate already gives F_i\le C_1. Since \beta is bounded away from zero on \mathcal K, this proves uniform second moments. A final Cauchy–Schwarz inequality proves the assertion.□

Proof. Differentiate the logarithm of the grand partition function along the interpolation. The absolute derivative is at most

Including particle self pairs only enlarges this bound. Differentiation under the integrals and sums is valid by the common negative quadratic in the finitely many cell counts. Proposition 3.1 and integration over the interpolation parameter prove (3.1).

For the tail difference, each row sum is at most c_b t\sum_{j>n}\sigma_j, and the number of cells is at most (L/b+2)^3. Divide (3.1) by tL^3, use the pressure limit in Proposition 2.3, and let L\to\infty. The bounds are uniform on \mathcal K, proving (3.2).□

Proof. Fix the reference potential, and let R_0 bound its interaction range. All constants depending on R_0 are fixed during this proof. Partition a cube of side L=mu into m^3 boxes of side

Translate the b-cell lattice, if necessary, to align it with the outside cube; all preceding estimates are invariant under this translation. The u-boxes are then unions of whole b-cells. For each R, we first let m\to\infty; only afterward do we let R\to\infty. In particular u\to\infty, u/R\to0, and eventually u\gg R_0.

Write B_z for the box centered at z, and define

Choose the fixed constant C large enough that K^-_{zz'} is at most the corresponding infimum. The C^1 bound on w and its compact support imply, for some \eta_R=O(u/R)\to0,

If z is farther than R+3u from the outside cube boundary, then

Here the sums are over the finite center grid in the cube. To check these estimates, compare u^3\sum_{z'}w_R(z-z') with \int w_R=1. The oscillation in each grid cube is O(uR^{-4}), and only O((R/u)^3) cubes meet the support enlarged by 3u. The total error after multiplication by u^3 is O(u/R). The same argument controls the suprema and the truncated lower kernel. Also K^\pm_{zz'}\le C R^{-3}.

Upper bound. Delete all positive core interactions between the u-boxes. This increases the partition function. Remove next the reference attractive interactions between these boxes, keeping the new Kac attraction in place. By Proposition 3.2, this second change costs at most C_{\mathcal K,R_0}L^3/u in the logarithm. Indeed only original cells within R_0+2b of a box boundary have a nonzero difference row, and there are O_{R_0}(L^3/u) such cells. Proposition 3.1 applies throughout this interpolation.

Write n_z for the count in the box centered at z. The long-range pair sum is at most

where the last step uses symmetry and n_zn_{z'}\le(n_z^2+n_{z'}^2)/2. The first bound includes self pairs, which is harmless for this upper estimate of an attractive Boltzmann factor. The remaining reference interactions are confined to individual boxes. Summing the assignments of particle labels cancels the global canonical factorial and gives

The finite-volume bound (2.12) and \kappa>10t(A+1) leave the effective energy coefficient

uniformly away from zero for all sufficiently large R. Consequently counts n/u^3>D, for sufficiently large D, are suppressed by a uniform negative quadratic. The sum of these high-count terms is exponentially small compared with the empty term. Among the remaining O(u^3) terms, every maximizing subsequence has a further subsequence with n/u^3\to d\in[0,D]. The all-sequence limit (2.7) then gives

The logarithm of the number of low-count terms divided by u^3 vanishes. This proves the equality by the preceding upper subsequence argument and by taking n=\lfloor du^3\rfloor for the lower bound. Taking first m\to\infty in (3.6) and then R\to\infty proves the desired upper estimate.

Lower bound. Place a cube of side s=u-2 at each of the same centers, and require exactly n=\lfloor ds^3\rfloor particles in every smaller cube, where d\ge0 is fixed. The gaps have width two. Thus all reference interactions between distinct smaller cubes are nonpositive and can be discarded in a lower bound on the partition function. Let m_{\mathrm{int}} be the number of centers satisfying the interior condition in (3.5). The Kac pair sum obeys

The last term removes the particle self pairs included in the diagonal box contribution. The count constraint and the canonical factorials therefore give

For fixed R, m_{\mathrm{int}}/m^3\to1 as m\to\infty. Next R\to\infty gives s/u\to1, n/s^3\to d, and a vanishing self-pair error. By (2.7), the resulting pressure lower bound is

Take the supremum over d\ge0.

We have proved pointwise convergence in (3.3). The functions on both sides are finite and convex on the open convex set \Omega, by Propositions 2.3 and 2.4. Pointwise convergence of finite convex functions to a finite convex function is uniform on compact subsets, which completes the proof.□

Proof. Call a configuration free if every pair distance below one lies in [e^{-t},(1+\delta)e^{-t}]. Join points at distance below one. A two-edge path has endpoint distance at most 2(1+\delta)e^{-t}<1; freeness makes the endpoints adjacent. Thus connected components are complete graphs. The geometric estimate in Lemma 2.1 bounds their sizes by four. Representatives from different components are separated by at least one, so their number is at most M_L.

Let V=L^3, and let \Xi_{\mathrm{free}} denote the grand configurational integral restricted to free configurations. Root each component and integrate its root over volume V. Every other point in the component lies in a ball of volume at most c_0e^{-3t} about the root. The labeled block enumeration, followed by division by the canonical factorial, gives the upper bound

For completeness, ordered component sizes give a factor 1/(k!\prod m_i!); allowing any of the m_i labels to be a root replaces m_i! by (m_i-1)! and only increases the count. Ignoring separation between the roots also enlarges the integral.

A bad edge is a pair at distance below one but outside the allowed shell. If there are B bad edges, their endpoint set S has size at most 2B, its complement is free, and the core energy is at least t^2B. Hence, pointwise on labeled configurations,

The term corresponding to the actual bad-edge endpoint set proves this inequality; zero Boltzmann factors cause no difficulty. Summing over particle numbers and integrating the removed particles gives

Now

The convergence is uniform for bounded h. For u=k/V, the factorial estimate k!\ge(k/e)^k bounds the remaining normalized logarithm in (4.2) by u(1-\log u)/t, with value zero at u=0. It is uniformly o(1) for bounded u as t\to\infty. There are only M_L+1=O(V) terms. Take L\to\infty first and use M_L/V\to p. The extra pressure in (4.3) is e^{th-\beta t^2/2}/t, which tends to zero uniformly on compact subsets of \Omega. This proves the upper limit in (4.1).

For the lower limit, fix \ell>0 and \eta>0. Dilate an optimal M_\ell-packing by 1+\eta, and repeat it with cubic period

In a cube of side mP, the k=m^3M_\ell resulting centers have mutual distances at least 1+\eta and a fixed positive margin from the outside boundary. At each center put either one particle or four particles, using the same choice m_0\in\{1,4\} everywhere. The root ranges over a ball of fixed volume v_0>0, small compared with the separation margin. For m_0=4, put the other particles in relative balls of volume v_1e^{-3t} about the three other vertices of a regular tetrahedron of side (1+\delta/2)e^{-t}. A sufficiently small fixed v_1>0 puts every internal distance in the shell. For all large t, distances between clusters exceed one.

All these configurations have zero core energy. Assign unordered sets of m_0 labels to the k centers and choose one deterministic ordering of each set for the root and vertices. Different assignments give disjoint integration regions, and the relative coordinates have Jacobian one. Thus

First let m\to\infty, then t\to\infty. The contribution to q_t has lower limit

The empty configuration supplies zero. Finally let \ell\to\infty and \eta\downarrow0. This proves the pointwise limit. Local uniform convergence follows again from finite convex convergence on \Omega.□

Proof. Applying the first inequality in (2.5) to the pure core gives (2.8) with \kappa=t^2b^3/8 and B_0=t^2/2. Consequently

For large t, the quadratic in this bound dominates the added term \beta Ad^2/2. On a fixed compact parameter set, the transformed objective is negative for d\ge D, with D independent of all large t. Since d=0 gives zero, all maximizers lie in [0,D].

The partial conjugate of q_* is

Consider a subsequence with d_j\to d. For every z\in\R, Fenchel’s inequality gives

Take the upper limit and then the infimum over z. The result is bounded by

In particular d\in[0,4p], since the transform is nonnegative. This objective is strictly convex on each of the two linearity intervals of H_*; its maximum is therefore the right side of (4.4).

For the matching lower bound, choose a maximizing d_0\in\{0,p,4p\} for this limiting objective. Its global optimality says that, for every d',

Thus z=h+\beta Ad_0 exposes the unique conjugate density d_0 for q_*. Set z_j=h_j+\beta_jAd_0, and choose a subgradient (s'_j,d'_j)\in\partial q_{t_j}(\beta_j,z_j). The physical pressure is nondecreasing in h, so d'_j\ge0; these are admissible densities for the transform. By Proposition 4.1 and (4.5), every cluster here is (0,d_0). Fenchel equality at d'_j therefore gives

The upper and lower limits agree. Equality in the upper argument forces every limiting maximizing density to maximize the limiting objective, and strict convexity on its two intervals confines such densities to \{0,p,4p\}. The argument allows moving parameters; it proves the assertions, including local uniform convergence.□

Proof. For e in the right side and every (\beta,h)\in\Omega, the defining conjugate inequality gives

This is the joint supporting inequality. Conversely, a support (e,d) gives

for every h. Take the supremum in h to obtain the supergradient inequality for H(\,\cdot\,,d).□

Proof. Compare the transformed objective at d with its value at d' and complete the square to obtain (4.8). Dropping the nonnegative quadratic shows that z supports H(\beta,\cdot) at d, so Fenchel equality holds. Lemma 4.3 gives the first inclusion in (4.9). For the second, the fixed-density branch

lies below F and agrees with it at (\beta,h). The canonical supergradient -e gives this branch the displayed supporting vector, which therefore supports F as well. At a differentiability point there is only one supporting vector.□

Proof. The function P_*(y)-g\cdot y has its unique minimum at \lambda_*. In fact

for some \gamma_0>0, because the triangle of the v_i contains a ball about g. Minimize P_1^t(y)-g\cdot y over \overline B(\lambda_*,r_0/16). Lemma 4.2 shows that all minimizers lie in the interior for large t and converge to \lambda_* as t\to\infty. Choose one and call it \lambda_1^t. Interior minimality gives g\in\partial P_1^t(\lambda_1^t). It remains to locate the slopes generating this subdifferential.

We claim that any limit of gradients \nabla P_1^{t_j}(\beta_j,h_j), at differentiability points with t_j\to\infty and (\beta_j,h_j)\to\lambda_*, belongs to \{v_1,v_2,v_3\}. Choose a maximizing density d_j at each such point. Apply Lemma 4.4 with Q=q_{t_j} and a=A, choosing e_j\in-\partial_\beta^+H_{t_j}(\beta_j,d_j). With z_j=h_j+\beta_jAd_j, it gives

and

The shifted support controls the original first-coordinate slope e_j before the explicit increment Ad_j^2/2 is added. The maximizing densities are bounded by Lemma 4.2; hence the z_j stay in a compact interval. The locally uniform limit q_* is independent of \beta, so every cluster of the subgradients in (4.14) has first coordinate zero. Therefore e_j\to0. The same lemma puts every density cluster in \{0,p,4p\}, and (4.13) proves the claim.

Subgradients of P_1^t near \lambda_* are bounded uniformly for large t, by (4.4). If G_{P_1^t}(\lambda_1^t) were not contained in \bigcup_i\overline B(v_i,\tau/2) for arbitrarily large t, choose such a sequence of reachable gradients. Approximate each by a gradient at a differentiability point within distance 1/t of \lambda_1^t, with gradient error tending to zero. A bounded subsequence then contradicts the preceding claim. Thus, for all sufficiently large t, the three intersections

cover the reachable-gradient set. Let K_i^1 be their convex hulls. They are compact and remain inside their respective balls. Formula (4.6) gives the equality in (4.12). None of the three sets is empty: otherwise g would lie in the convex hull of two of the larger balls U_i. Their density coordinates are nonnegative because P_1^t is nondecreasing in h.

Finally, choose one point in each K_i^1. Their triangle contains \overline B(g,\gamma) by the fixed geometry of the U_i, and is contained in \partial P_1^t(\lambda_1^t).□

Proof. We prove the conclusion along an arbitrary sequence a_m\downarrow0 and physical pressures Q_m with \|Q_m-P\|_W\leq a_m^2. The radius S will be chosen from the fixed limiting data. A contradiction argument will then supply one threshold a_* valid for all admissible Q.

The geometric effect of the quadratic. Write \xi=(x,y) and \pi_2(s,d)=d. Define the finite convex function

This is the function that will arise when the attractive mass and the distance to \lambda_0 tend to zero at the same scale. Before proving that limit, we identify which points of K can attain its maximum. Every maximizer of \xi\cdot(s,d)+\beta_0d^2/2 over K belongs to one of the K_i. Indeed, write such a maximizer as a finite convex combination v=\sum_j\theta_jv_j, with v_j=(s_j,d_j)\in\bigcup_iK_i and positive weights. If two of the d_j differ, strict convexity of the quadratic gives

contradicting maximality. All d_j must therefore coincide. The disjoint density projections put all v_j in one K_i, and convexity of that K_i gives v\in K_i. Thus the positive quadratic rules out a maximizing mixture across the three density groups. To use this observation for physical pressures, we must show that limits of their transformed gradients are themselves maximizing points of K; membership in the convex hull of such points would not suffice.

Step 1: the pressure at scale a_m. On the expanding open set \Omega_m=\{\xi\in\R^2:\lambda_0+a_m\xi\in\Omega\}, put

Every fixed compact set of scaled coordinates is mapped into \operatorname{int}W for all sufficiently large m. On that compact set, the difference between q_m and the corresponding rescaling of P is at most a_m. The rescaling of P converges to its directional derivative at \lambda_0, which is h_K. This convergence is uniform on compact sets: the rescaled functions have a common local Lipschitz bound, and their pointwise limits are the directional derivatives. Consequently

The scaling preserves subgradient vectors:

It follows that, whenever \xi_m\to\xi and v_m\in\partial Q_m(\lambda_0+a_m\xi_m), the vectors v_m are bounded and every limit belongs to F_\xi. To see the last assertion directly, pass the supporting inequalities of q_m to the limit. They give v\in\partial h_K(\xi)=F_\xi. This observation applies to faces of any dimension; no uniqueness of the supporting vector is assumed.

Step 2: the transform at the same scale. Put

We will prove that

To keep track of maximizing densities as well as function values, we use the slices of K. For fixed x\in\R, define

with value -\infty when d is outside the density projection of K. The function k_x is upper semicontinuous and concave on its compact interval of finite values. Partial conjugacy of the support function gives

Indeed h_K(x,z)=\sup_d(k_x(d)+zd); the reverse formula is concave biconjugacy, or equivalently separation of the closed hypograph of k_x.

We will also retain the following information about optimizers: if \xi_m=(x_m,y_m)\to(x,y) and d_m maximizes the transform at \lambda_0+a_m\xi_m, then every limit d of the d_m satisfies

Write \beta_m=\beta_0+a_mx_m and h_m=h_0+a_my_m. The physical uniform bound on optimizing densities makes (d_m) bounded. For every fixed z\in\R, Fenchel’s inequality at (\beta_m,h_0+a_mz) gives

This will yield the upper bound after taking a density limit and then the infimum over z.

For the lower bound, choose v_*=(s_*,d_*)\in K that attains the maximum defining r(x,y), and put z=y+\beta_0d_*. Comparing its value with that at any (s',d')\in K and completing the square gives

Thus v_*\in F_{(x,z)}, and every member of this face has density d_*. Choose a full subgradient of Q_m at (\beta_m,h_0+a_mz), and denote its density coordinate by d'_m. Step 1 shows that d'_m\to d_*; nondecreasing dependence on h gives d'_m\geq0. Fenchel equality at d'_m permits evaluation of the transformed objective and yields

Now pass to a subsequence in (5.11) along which d_m\to d. Taking the infimum over z after the limit and using (5.9) gives

The finite lower bound (5.13) excludes d\notin\pi_2K. Equality throughout proves (5.10). These arguments apply to every convergent sequence \xi_m in a fixed compact set. They therefore prove the uniform convergence (5.8) as well.

We have identified the exact effect of an attraction of strength a_m on the pressure at distance O(a_m) from \lambda_0. The positive quadratic in (5.5) is now visible at order one. We next use it to locate a nearby subdifferential containing g and to keep all of its reachable gradients within the three groups.

Step 3: a nearby point supporting g. By (5.2) and \beta_0>0,

Choose S>0 such that \gamma S>r(0)+1. Let \xi_m minimize B_m(\xi)-g\cdot\xi on the closed ball \overline B(0,S). Uniform convergence on this ball and (5.14) show that, for large m, all such minimizers are interior: every boundary value exceeds the value at 0. Every limit of these minimizers is also interior. Hence, at

convex minimality gives

For large m these points lie in \operatorname{int}W.

Step 4: localization of every reachable gradient. Put G_m=G_{\T_{a_m}Q_m}(\lambda'_m). We claim that, eventually,

Suppose otherwise, and choose u_m\in G_m outside this union along a subsequence. By the definition of a reachable gradient, there is a differentiability point \widehat\lambda_m of \T_{a_m}Q_m such that

Write \widehat\lambda_m=\lambda_0+a_m\widehat\xi_m, with \widehat\xi_m=(\widehat x_m,\widehat y_m). The scaled points remain in a fixed compact set. Moreover, scaling preserves the gradients of the transforms:

Their boundedness thus follows from the local uniform convex convergence in (5.8). After passing to a subsequence, assume

Let d_m be any maximizing density at \widehat\lambda_m=(\widehat\beta_m,\widehat h_m). Apply Lemma 4.4 with Q=Q_m and a=a_m. Because H_{Q_m}(\,\cdot\,,d_m) is finite and concave, we can choose e_m\in-\partial_\beta^+H_{Q_m}(\widehat\beta_m,d_m); the lemma gives

and

In particular d_m\to d, and the first-coordinate increment tends to zero. The argument on the right has scaled coordinates

Equations (5.17) and (5.18), together with Step 1, show that u\in F_\eta.

By (5.10), there is a point v_*=(s_*,d)\in K that maximizes the expression defining r(x,y). Equation (5.12) shows that v_*\in F_\eta as well. The vectors u and v_* have the same density and the same value against \eta, so xs=xs_*. Consequently u itself maximizes that expression. This conclusion also holds when x=0; an exposed face need not be a singleton.

Since u is a maximizer, the geometric observation at the start of the proof puts u in one of the K_i. This contradicts the choice of u_m outside the fixed \zeta/2 neighborhoods. This proves (5.16).

Step 5: the three groups and a uniform threshold. Define

Reachable-gradient sets are compact. Hence every nonempty J_{i,m} is compact and convex, and lies in K_i+\overline B(0,\zeta/2). The reachable-gradient identity and (5.16) give

None of the three sets is empty: otherwise (5.15) would put g in the convex hull of two of the allowed enlarged groups. The density coordinate of every reachable gradient is nonnegative because \T_{a_m}Q_m is nondecreasing in h. Thus the J_{i,m} have all the asserted properties.

We have proved the conclusion for all sufficiently large indices along every sequence a_m\downarrow0, with the same radius S. If no uniform threshold a_* existed, one could choose a sequence of counterexamples with a_m\downarrow0, contradicting what was just proved. Decrease a_* if necessary so that a_*\leq\min(1,A+1) and \lambda_0+\overline B(0,Sa_*)\subset\operatorname{int}W. This completes the proof.□

Proof. Choose a cube with center c whose concentric enlargement lies in O. Convexity bounds a function above on the larger cube by its values at the vertices. The inequality F(y)\ge 2F(c)-F(2c-y) then bounds it below on a smaller concentric cube. On a still smaller cube, extend any chord by a fixed positive distance in both directions while remaining in the bounded region. Monotonicity of one-dimensional convex secant slopes bounds the difference quotient on the original chord by its surrounding oscillation divided by that distance. This proves local Lipschitz continuity. In the sequence assertion, the finitely many vertex values and the center value are uniformly bounded by pointwise convergence, so the same argument gives a common local Lipschitz constant. A finite net of points on each compact set upgrades pointwise to uniform convergence. Passing convexity inequalities to limits proves convexity of F.

Here is also a direct supporting-plane construction. Separate a point strictly below the epigraph from that convex epigraph, let the point tend to (x,F(x)), and normalize the separating normals. Separation in finite dimensions follows by projecting an exterior point onto a closed convex set: if z is its nearest point, expansion of the squared distance along z+\theta(y-z) gives the separating inequality. One can restrict the epigraph to a closed neighborhood of x before applying this construction. A limiting normal has a nonzero vertical component: a horizontal supporting normal would separate x from an open neighborhood of x, which is impossible. Division by that component produces a local affine minorant through (x,F(x)). Convexity extends it to all of O by considering the initial portion of the segment from x to any other point. Thus \partial F(x) is nonempty.

If v\in\partial F(x) and x\pm re_j\in O, the supporting inequalities bound |v\cdot e_j| by the oscillation of F divided by r. Uniform oscillation on a surrounding compact set bounds all subgradients at points in a smaller compact set. Subdifferentials are closed by their defining inequalities, hence compact. The same bounds apply to F_n under local uniform convergence. Passing F_n(y)\ge F_n(x_n)+v_n\cdot(y-x_n) to a convergent subsequence proves the last assertion.□

Proof. At every interior point of the domain, a slope between the left and right convex derivatives gives an affine minorant. Hence u has at least one affine minorant \ell_0. To obtain enough such minorants, fix a point (d,a) strictly below the closed epigraph of u. Strict separation, by the projection argument in the preceding proof, yields an affine inequality

Necessarily B\ge0. If B>0, division gives an affine minorant of u whose value at d exceeds a. If B=0, add a sufficiently small positive multiple of the valid inequality y\ge\ell_0(d'); the resulting inequality has positive vertical coefficient and still strictly separates (d,a). This also gives an affine minorant exceeding a at d. Since every point below the epigraph is so separated, u is the supremum of its affine minorants, including at boundary and exterior points of its domain. Among affine minorants of slope h, the largest is d\mapsto hd-u^*(h), which proves biconjugacy. Rearranging the definition of u^* proves the equality criterion.

For the last assertion, put k_x(d)=\max\{xs:(s,d)\in K\}. Compactness and convexity of K make k_x upper semicontinuous and concave on its compact projection interval. Thus -k_x, extended by +\infty, satisfies the preceding hypotheses. The identity h_K(x,z)=\sup_d(k_x(d)+zd) and biconjugacy give the formula.□

Proof. On every coordinate line, a one-dimensional convex function has equal left and right derivatives except at countably many points: both derivatives are monotone, and each strict gap contains a distinct rational number. Fubini’s theorem therefore implies that both coordinate partial derivatives of F exist almost everywhere. At such a point, every subgradient has these two coordinates, so the subdifferential is a singleton, say \{v\}. This implies differentiability. Indeed, for small u choose v_u\in\partial F(x+u). Local boundedness and closedness of subgradients show v_u\to v, and the two supporting inequalities give

Their difference is o(|u|). Thus differentiability points have full measure, in particular they are dense.

Local boundedness and diagonal subsequences show that G_F(x) is nonempty and compact. Each of its elements is in \partial F(x) by the limiting support inequality. Conversely, take p\in\partial F(x) and a direction e. Choose differentiability points y_j=x+a_je+o(a_j), with a_j\downarrow0. This is possible by density, for example with error less than a_j^2. Adding the supporting inequalities at x and y_j gives

After passing to a bounded-gradient subsequence and dividing by a_j, one obtains v\in G_F(x) with v\cdot e\ge p\cdot e. Thus G_F(x) and \partial F(x) have the same support function. In the plane the convex hull of a compact set is compact: any convex combination can be reduced to three terms by eliminating an affine dependence among four or more points. Separation therefore proves \partial F(x)=\conv G_F(x).

Finally, the convex secant slope in the last assertion has a limit as a\downarrow0. Supporting inequalities at x give the lower bound v\cdot e for every v\in\partial F(x). Choose instead v_a\in\partial F(x+ae); the reverse supporting inequality bounds the same secant slope above by v_a\cdot e. A bounded subsequence of v_a converges to an element of \partial F(x), proving the claimed equality. The local Lipschitz constant for F gives a common Lipschitz constant in e for all sufficiently small rescalings on any fixed compact set. A finite-net argument then proves uniform convergence there.□