Results on polar bodies, forms and flows, plurisubharmonic potentials, the planar map and the realization: separation and polarity, strip bodies, the Pfaffian, Moser's theorem, potentials, the ball from a holomorphic zero, the slice estimate and the two theorems, with their assumptions

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• Symmetric Mahler inequality (OpenAI) — Theorem A. For every n >= 1 and every origin-symmetric convex body K in R^n, vol(K) vol(K°) >= 4^n/n!, with equality for the cube [-1,1]^n. Proved in the last lesson.
• Symplectic balls in symmetric polar products (OpenAI) — Theorem B. For n >= 2, an origin-symmetric convex body K and every 0 < c < 4 there is an injective symplectic map from the ball B^(2n)(c) into int K x int K°, a diffeomorphism onto an open subset. Proved in the last lesson.
• Separation — Lemma 1.2. A point x outside a nonempty compact convex set C satisfies <v,x> > h_C(v) for some vector v.
• Support functions and polarity — Lemma 1.3. For nonempty compact convex sets, C is contained in C' if and only if h_C <= h_C'; for convex bodies containing 0 in their interiors, the bipolar of C is C.
• Strip bodies — Lemma 1.4. If b_1, ..., b_m span R^n, the strip body K' of all x with |<b_j,x>| <= 1 for every j is an origin-symmetric convex body, the polar of the convex hull P of the points ±b_j, and (K')° = P.
• Finite polar approximation — Lemma 1.5. For an origin-symmetric convex body K and alpha > 0 there is a strip body K' of spanning vectors with K in K' in (1+alpha)K and (K')° in K°.
• The Pfaffian of a congruent matrix — Lemma 2.3. Pf(M^T A M) = det(M) Pf(A) for every real 2n x 2n matrix M and skew-symmetric A, and Pf(J) = (-1)^(n(n-1)/2).
• Symplectic maps preserve volume — Lemma 2.4. A 2n x 2n matrix M with M^T J M = J has determinant 1; an injective smooth map e on an open set U with e^* omega_0 = omega_0 has open image of the same volume; vol(B^(2n)(c)) = c^n/n!.
• Cartan's formula for closed forms — Lemma 2.5. If the two-form Omega is closed, then L_Y Omega = d(iota_Y Omega).
• Compactly supported flows — Lemma 2.6. A smooth time-dependent vector field Y_s on R^N vanishing outside a compact set C has a smooth flow phi_s, s in [0,1], by diffeomorphisms that fix every point outside C and map C onto C, with d/ds D phi_s = DY_s(phi_s) D phi_s.
• Derivative of pullbacks along a flow — Lemma 2.7. For two-forms Omega_s on an open V, smooth in (s,x), and such a flow with C in V: each phi_s maps V onto V, and d/ds (phi_s^* Omega_s) = phi_s^*(d/ds Omega_s + L_(Y_s) Omega_s).
• Moser's theorem — Theorem 2.8. If symplectic forms Omega_s on an open V, smooth in (s,x), satisfy d/ds Omega_s = d gamma_s with one-forms gamma_s, smooth in (s,x), vanishing outside a fixed compact C in V, then a diffeomorphism phi of V, the identity outside C, has phi^* Omega_1 = Omega_0.
• Symplectic forms from potentials — Lemma 3.4. If phi is plurisubharmonic, then omega = dd^c(|z|^2 + phi) is a symplectic form with omega(X, iX) >= |X|^2.
• Norms of holomorphic maps — Lemma 3.5. For holomorphic f = (f_1, ..., f_m) and tau = |f|^2, dd^c tau(X, iX) = |df(X)|^2 >= 0, so tau is plurisubharmonic, and log tau is plurisubharmonic where f has no zero.
• Convex functions of potentials — Lemma 3.6. A smooth function Gamma, convex and nondecreasing in each argument, of plurisubharmonic functions h_1, ..., h_N is plurisubharmonic.
• Smooth maximum — Lemma 3.7. rmax(h,p) = (h + p + rho(h-p))/2, with rho the convolution of |s| with an even bump supported in (-1,1), is smooth, convex, nondecreasing in each argument and equal to max(h,p) when |h-p| >= 1.
• Potentials along a real direction — Lemma 3.8. For b in R^n, ell(z) = sum b_j z_j and phi holomorphic on D, dd^c |phi o ell|^2 = |phi'(ell(z))|^2 d<b,u> ^ d<b,x> on the set where ell(z) lies in D.
• Radial potentials — Lemma 3.9. If Psi' > 0 and (t Psi'(t))' > 0 on [0, r^2], the map R(z) = sqrt(Psi'(|z|^2)) z satisfies R^* omega_0 = dd^c Psi(|z|^2) and maps B_r diffeomorphically onto the ball B^(2n)(pi r^2 Psi'(r^2)).
• A ball from a holomorphic zero (OpenAI) — Theorem 4.1. If a holomorphic f on an open set containing 0 has f^(-1)(0) = {0}, |f(z)| <= C_0 |z|^k near 0 and compact sublevel sets {|f|^2 <= s} for 0 < s < 1, then for every 0 < c < pi k the ball B^(2n)(c) embeds by a map E with E^* omega = omega_0 into V = {|f|^2 < 1}, omega = dd^c(|z|^2 + |f|^2).
• An arctangent series — Lemma 5.1. a(w) = sum (-1)^l w^(2l+1)/(2l+1) is holomorphic on the unit disc with a'(w) = 1/(1+w^2), Re a(re^(i theta)) = (1/2) arctan(2r cos theta/(1-r^2)), and a(w) = 0 only for w = 0.
• The planar conformal map — Theorem 5.2. F is holomorphic, bounded, odd and injective on the unit disc with wF'(w) = (8/pi^2) a(w) and F'(w) nonzero; its image D has horizontal slices (-V(t), V(t)) for t in (-1,1), and on each slice the real part increases strictly with the radius.
• The uniform slice estimate (OpenAI) — Proposition 6.1. The slice integrals J_k are smooth on D and odd in v with d_v J_k = k^2 |g|^(2k-2) |g'|^2 >= 0, and |J_k(v,t)|/k <= (pi/4)|g(v+it)|^(2k) + eps_k for k >= 2, with eps_k -> 0 depending only on F.
• An arctangent bound — Lemma 6.2. 1/arctan X - 2/pi <= 2/X for every X > 0.
• A Beta-type integral — Lemma 6.3. k times the integral of r^(2k-1) sqrt((1-r)/(r-r_a)) over (r_a, 1) is at most 1 + pi/e, for integers k >= 1 and 0 <= r_a < 1.
• Balls in polar products of strip bodies (OpenAI) — Proposition 7.1. For n >= 2, the strip body K of spanning vectors b_1, ..., b_m and every 0 < c < 4 there is an injective symplectic map from B^(2n)(c) into int K x int K°, a diffeomorphism onto an open subset.

Reading order

Read the lessons in order. The first lesson states both theorems and proves the facts on polar bodies; the next two set up forms, flows and plurisubharmonic potentials; Lessons 4 to 6 build a ball from a holomorphic zero and the planar map with its slice estimate; the last lesson realizes the domain in a polar product and proves Theorems B and A.