Results on projection bodies: Cauchy's formula, affine covariance, products, the simplex value and the counterexamples, 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.

• A product of two simplices beats the simplex (OpenAI) — Theorem A. R_20(T_10 x T_10)/c_20 = 121 C(20,10)/(21 2^20) = 22,355,476/22,020,096 > 1. Proved in the third lesson.
• Exponential excess (OpenAI) — Theorem B. There are lambda > 1 and N such that for every n >= N some product of simplices K_n in R^n has R_n(K_n) >= lambda^n c_n. Proved in the third lesson.
• Support functions determine compact convex sets — Lemma 1.1. For nonempty compact convex C, D: C is contained in D if and only if h_C <= h_D.
• Projection of a facet — Lemma 2.2. The (d-1)-volume of the orthogonal projection of a facet onto u-perp is its area times |<a_j, u>|.
• Cauchy's formula for polytopes — Proposition 2.3. For a polytope P and unit u, vol_(d-1)(projection of P onto u-perp) = (1/2) sum_j s_j |<a_j, u>|; the projection body is the zonotope of the segments [-s_j a_j/2, s_j a_j/2].
• Affine covariance (OpenAI) — Proposition 3.1. Pi(LP + x_0) = |det L| L^(-T) Pi P for invertible L, and R_d is invariant under invertible affine maps.
• Projection bodies of convex bodies (OpenAI) — Proposition 3.2. Every convex body K has a projection body Pi K; circumscribed polytopes P_m have R_d(P_m) tending to R_d(K), and R_d is affine invariant on convex bodies.
• Products (OpenAI) — Proposition 4.1. For polytopes A in R^r and B in R^s, r, s >= 2: Pi(A x B) = (|B| Pi A) x (|A| Pi B) and R_(r+s)(A x B) = R_r(A) R_s(B).
• Facets of the simplex — Lemma 1.2. The standard simplex T_d has facet vectors -e_i/(d-1)! and (e_1 + ... + e_d)/(d-1)!.
• Extrusion — Lemma 2.1. For compact convex C and v != 0, |C + [0, v]| = |C| + |v| vol_(d-1)(projection of C onto v-perp).
• The simplex value (OpenAI) — Proposition 2.2. |Pi T_d| = (d+1)/((d-1)!)^d and R_d(S) = c_d = (d+1) d^d/d! for every d-dimensional simplex S.
• The product test (OpenAI) — Proposition 2.1. R_(r+s)(T_r x T_s)/c_(r+s) = ((r+1)(s+1)/(r+s+1)) C(r+s, r) r^r s^s/(r+s)^(r+s).
• The first dimension is twenty — Proposition 2.2. No product of simplices of dimensions at least two exceeds the simplex value in dimensions up to nineteen; T_10 x T_10 does in dimension twenty.

Reading order

Read the lessons in order: the first sets up projection bodies of polytopes and proves the product formula, the second computes the simplex value, and the third combines them. Section 3 of the first lesson, on general convex bodies, can be skipped by readers interested only in polytopes.