An exact quantitative shadow-system model

Definitions and mathematical statement

S. Reisner's source problem asks to quantify the shadow movement theorem, citing [MR], without specifying a deficit, distance, normalization or remainder. In the Meyer--Reisner interpretation used by the attached aid, a shadow system moves generating points linearly in one fixed direction and takes their convex hull; the relevant convex functional is inverse Santaló-polar volume, not the volume product itself.

The aid supplies the planar benchmark \[ K_t=\operatorname{conv}\{\pm(1+t,1),\ \pm(0,1),\ \pm(-1,0)\},\qquad-1<t<1. \] Only the first opposite pair moves, in the horizontal direction with scalar speeds $1,-1$; thus maximum generator speed is one. Central symmetry makes its Santaló point the origin, so its Santaló polar is the usual $K_t^\circ$. The exact areas and functional are \[ |K_t|=3+t,\qquad |K_t^\circ|=3-t,\qquad F(t)=|K_t^\circ|^{-1}=\frac1{3-t}. \] For $s,u\in(-1,1)$, $0\le\lambda\le1$ and $m_\lambda=(1-\lambda)s+\lambda u$, define the Jensen deficit $J=(1-\lambda)F(s)+\lambda F(u)-F(m_\lambda)$. The aid states \[ J=\frac{\lambda(1-\lambda)(u-s)^2}{(3-s)(3-u)(3-m_\lambda)} \ge\frac{\lambda(1-\lambda)(u-s)^2}{64}. \] In particular the midpoint deficit is at least $(u-s)^2/256$, with zero at coincident parameters or endpoint weights. The contrasting volume product is $|K_t||K_t^\circ|=9-t^2$, concave in $t$, so it must not be substituted for the inverse-polar functional.

The aid also notes that parameter rescaling changes any parameter-based quadratic coefficient, translations may give genuine motion with zero deficit, and affine Meyer--Reisner equality families supply further null directions. A general positive quantitative remainder must therefore state speed/scale normalization and how affine null directions are removed. The planar formula is a test case, not a quantitative theorem for every shadow system.

Short English statement

Quantify shadow-system convexity with a specified normalization and treatment of affine equality directions. The supplied aid gives an exact unit-generator-speed hexagon deficit and warns that the volume product is a different, here concave, functional.

Sources

• Local UnsolvedMath registry, permanent ID $20001286$, source code catalogue entry; canonical aim-convex-geometry-notes.json, notes[17].
• American Institute of Mathematics, "Mahler's conjecture and duality in convex geometry," stored problem 2 attributed to S. Reisner: https://aimath.org/WWN/mahlerduality/mahlerduality.pdf. The source cites [MR] without a bibliographic entry; a terminal page-number spill is not part of the question. No PDF inspection is claimed here.
• Attached research attempt 1: partial result, recorded run status open, rigor proved, novelty confidence low; registry status partially solved, classification PARTIAL-PROGRESS. It identifies the Meyer--Reisner functional and supplies the model computation but no full theorem citation. This machine-generated, non-peer-reviewed aid requires independent verification before citation.