--- title: Mixed Christoffel-Minkowski Problem url: https://www.emergentmind.com/topics/mixed-christoffel-minkowski-problem type: topic --- # Mixed Christoffel-Minkowski Problem The mixed Christoffel–Minkowski problem denotes a family of inverse problems in convex geometry in which one prescribes a mixed area measure, or an equivalent curvature density on the sphere, and seeks a convex body realizing that datum. In the formulations represented by recent work, the problem includes prescribing \(S(K_1,\dots,K_{n-1};\cdot)\) with all but one reference body fixed, prescribing repeated mixed area measures \(S_k(K;L_{k+1},\dots,L_{n-1};\cdot)\), and solving support-function equations involving combinations of the elementary symmetric functions of \(\nabla_S^2u+uI\) [2508.09800, 2508.09794, 2307.06252]. The subject links Brunn–Minkowski theory, elliptic and parabolic PDE, mixed volumes, and rigidity theory; the central questions are existence, uniqueness, regularity, and whether an analytic solution is geometric, i.e. whether it is the support function of a convex body. ## 1. Geometric formulations and basic objects A standard formulation fixes convex bodies \(K_1,\dots,K_{n-1}\subset\mathbb R^n\) and considers the mixed area measure \[ S(K_1,\dots,K_{n-1};\beta),\qquad \beta\subset S^{n-1}\ \text{Borel}, \] characterized by the polarization identity \[ V(L,K_1,\dots,K_{n-1}) = \frac1n\int_{S^{n-1}} h_L(u)\,S(K_1,\dots,K_{n-1};du) \] for every convex body \(L\). In the anisotropic Christoffel problem one fixes \(C\in\mathcal K(\mathbb R^n)\) and asks for necessary and sufficient conditions on a finite Borel measure \(\mu\) so that \[ \mu=S_1(K,C;\cdot) \] for some convex body \(K\) [2508.09800]. A closely related notation repeats one free body several times. For \(1\le k\le n-1\) and fixed bodies \(L_{k+1},\dots,L_{n-1}\), one writes \[ S_k(K;L_{k+1},\dots,L_{n-1};\cdot) := S(K,\dots,K,L_{k+1},\dots,L_{n-1};\cdot), \] with \(K\) appearing \(k\) times. The corresponding problem is to characterize those Borel measures \(\mu\) that can be represented as \[ \mu=S_k(K;L_{k+1},\dots,L_{n-1};\cdot) \] for some convex \(K\) [2508.09794]. In support-function language, if \(K\subset\mathbb R^{n+1}\) is smooth and strictly convex, its support function is \[ h(x)=\max_{y\in K}\langle x,y\rangle,\qquad x\in S^n, \] and the matrix \[ \nabla_S^2 h+hI \] has eigenvalues equal to the principal radii of curvature. Writing \(\sigma_k\) for the \(k\)-th elementary symmetric function, the classical \(k\)-th Christoffel–Minkowski problem prescribes \(\sigma_k(\nabla_S^2h+hI)\), while mixed variants prescribe combinations of the \(\sigma_\ell\) or more general weighted measures [1909.03645]. The literature represented here uses the phrase “mixed Christoffel–Minkowski problem” in several adjacent senses. One sense is the mixed area-measure problem with fixed reference bodies [2508.09800, 2508.09794, 2512.08670]. Another is the prescribed mixed curvature equation \[ \sum_{\ell=0}^k c_\ell\,dS_\ell = f(x)\,dS^n(x),\qquad c_\ell\ge0, \] which becomes a support-function PDE [1909.03645]. A third consists of \(L_p\), dual, and \((p,q)\) variants that mix area measures with dual curvature measures or related geometric measures [2010.11373, 2203.05105, 2504.04931]. ## 2. Support-function equations and PDE reductions On the sphere, a function \(u:S^n\to\mathbb R\) determines the hypersurface \[ X(x)=\nabla_S u(x)+u(x)x, \] and the spherical Hessian \[ W(u):=\nabla_S^2u+uI \] encodes the principal radii of curvature. Guan and Zhang study the general equation \[ \sigma_k(W(u))+\alpha(x)\sigma_{k-1}(W(u)) = \sum_{\ell=0}^{k-2}\alpha_\ell(x)\sigma_\ell(W(u)), \] with \(\alpha>0\) and \(\alpha_\ell\ge0\). When \(\alpha\equiv0\) and \(\alpha_\ell\equiv c_\ell\), this becomes the mixed Christoffel–Minkowski equation \[ \sigma_k(W(u))+\sum_{i=1}^{k-1}c_i\,\sigma_i(W(u))=\phi(x),\qquad c_i\ge0, \] which prescribes a convex combination of area measures [1909.03645]. Ivaki studies a non-homogeneous isotropic mixed Christoffel–Minkowski type operator \[ \mathcal C[u](x):=\sum_{k=0}^n \alpha_k\,\sigma_k\!\bigl(\bar\nabla^2u(x)+u(x)\bar g\bigr), \] where \(\alpha_0,\dots,\alpha_n\ge0\) and at least two coefficients are strictly positive. The prescribed equation is \[ \sum_{k=0}^n \alpha_k\,\sigma_k(\bar\nabla^2u+u\bar g) = \psi(u,|\bar\nabla u|)\,f(x). \] When \(f\equiv1\) and \(\psi\equiv1\), this is exactly the classical mixed Christoffel–Minkowski problem posed by Firey and Schneider in the 1970’s [2307.06252]. For fixed reference bodies of class \(C^{2,+}\), the mixed area-measure problem can also reduce to a linear elliptic PDE. If \(\Omega_1,\dots,\Omega_{n-1}\) are fixed and \(u\) is the support function of the unknown body \(\Omega\), then the mixed area density is \[ dS(\Omega_1,\dots,\Omega_{n-1},\Omega;\theta) = \mathscr D(W_1(\theta),\dots,W_{n-1}(\theta),W(\theta))\,d\theta, \] where \(\mathscr D\) is the mixed discriminant and \(W_i=u_{i,ab}+u_i\delta_{ab}\). By linearity in the last slot there is a positive-definite coefficient matrix \(A(\theta)\) such that \[ \sum_{a,b=1}^n A^{ab}(\theta)\,\bigl(u_{ab}(\theta)+u(\theta)\delta_{ab}\bigr)=f(\theta). \] Thus the mixed Christoffel problem becomes a linear uniformly elliptic equation on \(S^n\), with the geometric issue shifted to proving that \(W(u)>0\) [2512.08670]. These PDE reductions clarify an important structural distinction. In the fixed-reference-body formulation, the unknown support function enters linearly through the mixed discriminant polarization [2512.08670]. In isotropic or mixed-curvature prescriptions, the equation is fully nonlinear in the eigenvalues of \(W(u)\) [2307.06252, 1909.03645]. ## 3. Existence, admissibility, and regularity For the fully nonlinear curvature equation on \(S^n\), Guan and Zhang prove a global existence theorem under group invariance. If \(0<\alpha_\ell\in C^{m,1}(S^n)\), \(\alpha\in C^{m,1}(S^n)\), and a group \(G\subset O(n+1)\) acts without fixed points with \(\alpha,\alpha_\ell\) \(G\)-invariant, then there exists an admissible solution \[ u\in C^{m+2,\gamma}(S^n),\qquad 0<\gamma<1, \] of the curvature equation, unique up to adding first spherical harmonics. The proof combines ellipticity and concavity of \[ G(\lambda)=\sigma_k(\lambda)-\sum_\ell \alpha_\ell(x)\sigma_\ell(\lambda) \] in \(\Gamma_{k-1}\), \(C^0\) and \(C^2\) estimates, Evans–Krylov, and a degree-theoretic homotopy argument [1909.03645]. For the fixed-body mixed Christoffel problem in the \(C^{2,+}\) category, Colesanti–Focardi–Guan–Salani first solve the linear elliptic equation \[ \sum_{a,b=1}^n A^{ab}(\theta)\,\bigl(u_{ab}(\theta)+u(\theta)\delta_{ab}\bigr)=f(\theta) \] modulo linear functions, assuming \[ f\in C^{2,\gamma}(S^n),\quad f>0,\quad \int_{S^n}\theta\,f(\theta)\,d\theta=0. \] They then prove a constant-rank theorem: if \(A(\theta)\) is positive-definite and the mixed concavity condition (3.7) holds, any \(C^2\) solution with \(W(\theta)\ge0\) has constant rank, and on the full sphere this forces rank \(n\), hence \(W(\theta)>0\). Under the additional condition that the \(1\)-homogeneous extension of \(1/f\) to \(\mathbb R^{n+1}\) is convex, Theorem 3.4 yields a unique, up to translation, \(C^{2,+}\) convex body solving the mixed problem [2512.08670]. The role of admissibility is central throughout. In Guan–Zhang, admissibility means \(W(u)\in\Gamma_{k-1}\), which is enough for ellipticity and concavity of the operator but does not automatically imply full convexity \(W(u)\in\Gamma_n\). The paper explicitly identifies the extension of constant-rank arguments from the homogeneous \(\sigma_k\) case to inhomogeneous sums \(\sum c_\ell\sigma_\ell\) as an open but promising problem for obtaining genuine convex solutions of the mixed Christoffel–Minkowski problem [1909.03645]. Regularity theory depends on the formulation. In the global sphere problem, \(C^{2,\gamma}\) follows from uniform ellipticity and concavity, and higher regularity follows from Schauder theory [1909.03645]. In the linear mixed-Christoffel reduction, \(u\in C^{4,\gamma}(S^n)\) follows from standard elliptic theory once \(A^{ab},f\in C^{2,\gamma}\) [2512.08670]. In weak and non-smooth symmetry-reduced settings, the natural output is instead convex Aleksandrov solutions or continuous support functions, with smoothness neither assumed nor required [2508.09794, 2508.11600]. ## 4. Uniqueness, rigidity, and the Firey question A major rigidity result is Ivaki’s even isotropic uniqueness theorem. Let \[ \sum_{k=0}^n \alpha_k\,\sigma_k(\bar\nabla^2u+u\bar g) = \varphi(u,|\bar\nabla u|), \] where \(\varphi:(0,\infty)\times(0,\infty)\to(0,\infty)\) is \(C^2\), at least two \(\alpha_k\) are positive, and \(u\in C^2(S^n)\) is even and strictly convex. If \[ \ell^{-1}+u\,\partial_1(\log\varphi)(u,|\bar\nabla u|)>0 \] and \[ \partial_2\varphi(u,|\bar\nabla u|)\ge0, \] where \(\ell=\min\{k:\alpha_k>0\}\), then \(u\) is constant, hence the solution is the support function of an origin-centred sphere [2307.06252]. In the special isotropic case \(f\equiv1\) and \(\psi\equiv1\), this answers Firey’s 1974 question in the even isotropic regime: if a nonnegative linear combination of isotropic kinematic measures is proportional to surface area measure, then among origin-centred even strictly convex bodies the only solution is the sphere [2307.06252]. The proof uses the Heintze–Karcher inequality. For a closed convex hypersurface \(M\subset\mathbb R^{n+1}\) with mean curvature \(H\), \[ \int_M \frac1H\,d\mu_M \ge (n+1)V(\text{enclosed body}), \] with equality if and only if \(M\) is a round sphere. In support-function form, testing the equation \[ \varphi(h,|\bar\nabla h|)\,K=1 \] against \(\Delta h\) and integrating by parts, together with the sign conditions on \(\varphi\), forces equality in Heintze–Karcher and therefore spherical rigidity [2307.06252]. Immediate corollaries include uniqueness for the isotropic Orlicz–Minkowski problem under \(\partial_1\varphi\ge0\), \(\partial_2\varphi\ge0\), with at least one strict inequality, and for the isotropic \(L_p\)-Gaussian–Minkowski problem \[ h^{p-1}e^{2|\bar\nabla h|^2}K=c,\qquad p\ge1, \] for which every smooth strictly convex solution is a sphere [2307.06252]. The uniqueness statements in the broader literature are more varied. For fixed reference bodies, uniqueness is often only up to translation, reflecting invariance of area measures under translation [2508.09800, 2512.08670]. In some \(L_p\) or \((p,q)\) settings, uniqueness is up to dilation when \(p=q\) [2010.11373]. The literature therefore distinguishes translation-invariant inverse problems from rigidity results that force the recovered body to be spherical under isotropic and symmetry assumptions. ## 5. Symmetry-reduced complete solutions A complete solution is available in several symmetry classes. In the disk-area-measure problem, Brauner–Hofstätter–Ortega-Moreno fix the \((n-1)\)-dimensional disk \[ D=\{x\in e_n^\perp:\|x\|\le1\} \] and study measures of the form \(S_1(K,D;\cdot)\). If \(\mu\) is a nonnegative, centered, finite Borel measure on \(S^{n-1}\) with \(\mu(\{\pm e_n\})=0\), then there exists \(K\in\mathcal K(\mathbb R^n)\) with \[ \mu=S_1(K,D;\cdot) \] if and only if: \(\pi_*\mu\) is absolutely continuous with respect to rotation-invariant measure on the Grassmannian with continuous density; for almost every \(E\), the conditional measure \(\mu_E\) on \(S^1(E)\) is centered; and there exists a continuous support function \(h\) satisfying the integral equation \[ h(u)-\frac1\pi\int_{S^1(E_u)} h(v)(u\cdot v)\,dv = \int_{S^1(E_u)}\sqrt{1-(u\cdot v)^2}\,(\pi-\arccos(u\cdot v))\,\mu_{E_u}(dv). \] When these conditions hold, \(K\) is unique up to translation. The proof reduces the \(n\)-dimensional problem to planar Christoffel problems on the circles \(S^1(E)\) by disintegration and a Kubota-type projection formula [2508.09800]. Under axial symmetry, Brauner–Hofstätter–Ortega-Moreno obtain a complete solution without assuming regularity. If all bodies involved are symmetric about a common axis \(e_n\), then for a centered zonal measure \(\mu\) the existence of a body of revolution \(K\) with \[ \mu=S_i(K,C_1,\dots,C_{n-i-1};\cdot) \] is equivalent to five conditions: a support condition, non-degeneracy, positivity of a transferred measure on the interval \((a_-,a_+)\), finiteness of two boundary ratios, and an equator-mass condition. In that case the solution is unique up to vertical translation. The argument uses a one-dimensional integral operator \(T_R\) associated with the profile functions \(R_C\), the semigroup identity \(T_R\circ T_Q=T_{RQ}\), inversion of the adjoint transform, and reconstruction from planar sections [2508.09794]. In a more explicit rotationally symmetric setting, Mussnig–Ulivelli characterize the classical \(j\)-th area-measure problem for bodies of revolution by moment quotients \[ F_\mu^\pm(\alpha) = \frac{\int_{C_\alpha^\pm}|z_{n+1}|\,d\mu(z)}{\sin(\alpha)^{\,n-j}}. \] For a finite, centered, \(G\)-invariant Borel measure \(\mu\), there exists a body of revolution \(K\) with \(S_j(K,\cdot)=\mu\) and \(\mathrm{proj}_{e_{n+1}^\perp}K\) of positive radius if and only if \(F_\mu^+\) and \(F_\mu^-\) are non-trivial and non-decreasing. The associated radial convex functions are given explicitly by \[ u^\pm(x)=\int_0^{|x|}\Bigl(\frac1{\kappa_n}F_\mu^\pm(\arctan t)\Bigr)^{1/j}dt, \] and, after convex conjugation and gluing, one obtains an explicit three-piece formula for the support function on the sphere [2508.11600]. These symmetry reductions show that the mixed Christoffel–Minkowski problem can change character dramatically under invariance assumptions. In the general case, the problem may require nonlinear elliptic estimates or geometric admissibility theorems; in the rotationally symmetric case, it can reduce to one-dimensional monotonicity conditions or to compatible families of planar Christoffel problems [2508.09800, 2508.09794, 2508.11600]. ## 6. \((p,q)\), dual, and variational extensions Several recent works extend the mixed Christoffel–Minkowski problem beyond mixed area measures. Chen–Cui–Zhao introduce the \(k\)-th \((p,q)\)-mixed curvature measure \[ d\mu_k^{(p,q)}(K,x) = h_K(x)^{-p}\,\rho_K(x)^{\,q-n}\,\sigma_k\bigl(\nabla_{ij}h_K+h_K\delta_{ij}\bigr)\,dS(x), \] and pose the problem of finding \(K\in\mathcal K_o\) such that \(\mu=\mu_k^{(p,q)}(K,\cdot)\). For a smooth positive density \(f\), the equation is \[ h^{-p}\rho^{\,q-n}\sigma_k(\nabla_{ij}h+h\delta_{ij})=f(x). \] When \(q0\) even, and \[ \nabla^2\bigl(f^{-\frac1{k+p-1}}\bigr)+f^{-\frac1{k+p-1}}I\ge0, \] there exists at least one even, smooth, strictly convex solution. The proof combines \(C^0\), gradient, non-collapse, full-rank, and \(C^2\) estimates with Leray–Schauder degree theory. In the isotropic case \(f\equiv1\), the paper also proves spherical uniqueness under additional relations between \(p,q,k,n\) [2504.04931]. These extensions preserve the basic Christoffel–Minkowski pattern—recovering a convex body from a prescribed spherical measure—but replace the classical area measure by mixed, dual, or \(L_p\)-weighted analogues. They also show that no single method dominates the field: degree theory, curvature flows, variational maximization, constant-rank arguments, Kubota-type formulas, and sharp inequalities all appear as primary tools, depending on the geometric measure being prescribed [1909.03645, 2203.05105, 2010.11373, 2504.04931]. Source: https://www.emergentmind.com/topics/mixed-christoffel-minkowski-problem