% Unrefereed preprint.
% Copyright 2026 Ricardo Pinho. CC BY 4.0.
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\title[A cube--ball lower bound]{Optimizing a cube--ball lower bound for the reverse Brunn--Minkowski constant}
\author{Ricardo Pinho}
\address{Independent researcher}
\thanks{ORCID: \href{https://orcid.org/0009-0006-2744-2599}{0009-0006-2744-2599}. Licensed under CC BY 4.0.}
\date{23 October 2026; unrefereed preprint}
\keywords{reverse Brunn--Minkowski inequality, intrinsic volumes, exact certificates}
\begin{document}
\begin{abstract}
We optimize a single term in the cube--Euclidean-ball construction of
V. Patankar to obtain $C_{70}\ge \exp(1/(2e))>1.20194336847$.
The strongest public proposal located and reproduced in our dated source
audit is Patankar's $1.201927$, whose specified witness evaluates near
$1.2019271425725$. A separate finite-dimensional witness certifies
$1.20194336$. Two separately written arithmetic implementations, using
Arb and rational logarithm/Stirling bounds respectively, verify the
numerical comparisons and reject corrupted or excessive claims.
The geometric reduction and passage to the limit are proved here.
This unrefereed preprint is not peer reviewed and not formally verified;
exhaustive priority is not asserted.
\end{abstract}
\maketitle

\section{Statement and comparison}
Let $C_{70}$ be the infimum of $C$ such that for every pair of centrally
symmetric convex bodies $K,L\subset\R^n$, in every dimension, there is a
linear map $u$ with $|\det u|=1$ and
\[
 |uK+L|^{1/n}\le C\bigl(|K|^{1/n}+|L|^{1/n}\bigr).
\]
This is Tao and contributors' entry 70a~\cite{table}.
The existence of a finite constant is Milman's theorem; we address a lower
bound, uniformly against every permitted relative position.
\begin{theorem}\label{main}
One has
\[
 C_{70}\ge \exp\!\left(\frac1{2e}\right)>\frac{120194336847}{10^{11}}
 =1.20194336847.
\]
An explicit finite witness also gives
$C_{70}>3756073/3125000=1.20194336$.
\end{theorem}

The table's merged lower bound is merely $1$. The comparison relevant to
this note is the stronger open proposal of Patankar~\cite{patankar},
$1.201927$, from dimension $300000$ and cube side/ball radius $13/1000$.
We reproduced his unchanged public interval script before searching, and
an independently implemented Arb recurrence encloses its value near
$1.20192714257250284496098903$. Toledo also publicly replayed this
witness~\cite{toledo}. The new printed lower bound exceeds that witness
by more than $0.000016225897$. It is a modest optimization of the same
construction, not a new class of convex bodies.

\section{The positioning reduction}
Write $\kappa_j=\pi^{j/2}/\Gamma(1+j/2)$, with $\kappa_0=1$.
Take $K=[-a/2,a/2]^n$ and $L=B_2^n$.
\begin{lemma}[Patankar's cube--ball criterion]\label{criterion}
For every $a>0$ and every $n\ge1$,
\[
 C_{70}\ge
 \frac{\left(\sum_{k=0}^n\binom nk a^{n-k}\kappa_k\right)^{1/n}}
 {a+\kappa_n^{1/n}}.
\]
\end{lemma}
\begin{proof}
The parallelotope $P=uK$ is generated by vectors $v_1,\ldots,v_n$ with
volume $a^n$. Let $\delta_i>0$ be their squared Gram--Schmidt lengths.
For any $j$-element subset $S$, the squared volume of its spanned
parallelotope is at least $\prod_{i\in S}\delta_i$: projecting against
only earlier selected vectors cannot reduce length more than projecting
against all earlier vectors. Multiplying over $S$ and using
$\prod_i\delta_i=a^{2n}$ gives the principal-minor product inequality
usually attributed to Sz\'asz. AM--GM now gives
\[
 V_j(P)=\sum_{|S|=j}\operatorname{vol}_j(v_i:i\in S)
 \ge\binom nj a^j.
\]
Here $V_j$ are intrinsic volumes, with the usual zonotope normalization.
Steiner's formula therefore yields
\[
 |P+B_2^n|=\sum_{j=0}^n V_j(P)\kappa_{n-j}
 \ge\sum_{k=0}^n\binom nk a^{n-k}\kappa_k.
\]
This holds for every determinant-one-in-absolute-value $u$, while the
input root volumes are $a$ and $\kappa_n^{1/n}$. Hence every admissible
universal constant is at least the displayed ratio. Orthogonal equal-length
generators attain equality in this positioning comparison. This is the
existing construction and reduction of~\cite{patankar}.
\end{proof}

\section{One term and its limit}
For even integers $0<k<n$, put $t=k/n$ and set
\[
 a=\frac{1-t}{t}\,\kappa_n^{1/n}.
\]
Retaining just term $k$ in Lemma~\ref{criterion} proves $C_{70}\ge R_{n,k}$,
where
\begin{align}
 \log R_{n,k}
 &=\frac{\log\binom nk}{n}+(1-t)\log(1-t)+t\log t
     +\frac{\log\kappa_k-t\log\kappa_n}{n}\label{ratio}\\
 &=\frac{\log(n!)-\log(k!)-\log((n-k)!)
       -\log((k/2)!)+t\log((n/2)!)}{n}\notag\\
 &\hspace{18mm}+(1-t)\log(1-t)+t\log t.\label{factorial}
\end{align}
All powers of $\pi$ cancel in the second expression.
The chosen scale maximizes this single normalized term: differentiation
of $(n-k)\log s-n\log(1+s)$ gives $s=(n-k)/k$.

Fix $0<\theta<1$, let even $n\to\infty$, and choose even $k$ with
$k/n\to\theta$. Stirling's formula applied to~\eqref{factorial} gives
\[
 \frac{\log\binom nk}{n}
 =-t\log t-(1-t)\log(1-t)+O(\log n/n),
\]
and
\[
 \frac{-\log((k/2)!)+t\log((n/2)!)}n
 =-\frac t2\log t+O(\log n/n).
\]
The remainders are uniform when $t$ stays in a compact subinterval of
$(0,1)$. Thus $R_{n,k}\to\exp(-\theta\log\theta/2)$.
Taking $\theta=e^{-1}$ gives the first inequality in Theorem~\ref{main}.
Every finite witness bounds the same universal $C_{70}$ from below, so
passing to the limit requires no monotonicity assertion and no numerical
extrapolation. The derivative of $-\theta\log\theta/2$ vanishes at
$e^{-1}$, its second derivative is negative, and its limiting values at
the endpoints are zero. This optimizes this limiting single-term criterion;
no optimality over all convex-body pairs is claimed.

\section{Certificate and arithmetic verification}
The small certificate \texttt{certificates/c70\_candidate.json} specifies
\[
 n=10^{12},\quad k=367879441172,\quad
 a/\kappa_n^{1/n}=632120558828/367879441172.
\]
The finite expression~\eqref{factorial} gives
\[
 R_{n,k}=1.2019433684431505669948027797355\ldots>1.20194336.
\]
The first checker evaluates factorial logarithms and ball volumes with
256-bit Arb arithmetic. The second uses only rational arithmetic:
range-reduced
\[
 \log x=2\sum_{j=0}^{N-1}\frac{z^{2j+1}}{2j+1}+E,
 \quad z=\frac{x-1}{x+1},\quad
 0\le E\le\frac{2z^{2N+1}}{(2N+1)(1-z^2)}
\]
for $1\le x\le2$, with $N=44$, and Robbins's factorial inequalities
\[
 (m+\tfrac12)\log m-m+\tfrac12\log(2\pi)+\frac1{12m+1}
 <\log(m!)<
 (m+\tfrac12)\log m-m+\tfrac12\log(2\pi)+\frac1{12m}.
\]
Even the coarse bound $3<\pi<7/2$ suffices for its finite comparison.
The limit's decimal lower endpoint is separately checked using Arb and,
in the second implementation, a rational Taylor enclosure of $e$ through
term 60 followed by a rational logarithm comparison. This verifies the
number, while the preceding section supplies the limiting theorem.

Run, from the accompanying source directory,
\begin{verbatim}
python verify_c70_arb.py certificates/c70_candidate.json
python independent/verify_c70_rational.py \
       certificates/c70_candidate.json
python rejection_controls.py
\end{verbatim}
The controls include an excessive finite bound, an excessive limiting
bound, incorrect scale, odd dimensions, invalid schema and a valid but
insufficient half-dimensional term. The two programs share no search or
arithmetic code and use different libraries and formulas. Both were
authored by the same AI agent; this is algorithmic cross-checking, not
independent human authorship or a proof-assistant development.

\section*{Provenance and limitations}
The dated, bounded public-source audit is in \texttt{PRIOR\_ART.md} and
\texttt{FINAL\_LIVE\_CHECK.md}. Patankar's construction and positioning
argument are credited above; the present contribution is its optimized
single-term limit and a small rigorous certificate. No stronger C70 bound
was located in the sources checked; exhaustive priority is not asserted.
This AI-prepared preprint is not peer reviewed and not formally verified.

Analysis and proof drafting by AI agents under my direction; I reviewed
the results at summary level. This does not claim line-by-line human
verification.

\begin{thebibliography}{9}
\bibitem{table} T. Tao and contributors, \emph{Optimization Constants in
Mathematics}, entry 70a, \url{https://teorth.github.io/optimizationproblems/constants/70a.html}.
\bibitem{patankar} V. Patankar, \emph{Rigorous lower bounds for constants
16a, 20a, 20c and 70a, and an upper bound for 23b}, public code and proofs,
2026, source tree \texttt{c7c811aa06430527cccd5f992e41cd38dd9548ca},
\url{https://github.com/vidigoat/optimization-records}, Tao-table PR226.
\bibitem{toledo} C. Toledo, \emph{Five pull requests to the constants
registry, decided where decidable}, replay dated 16 October 2026,
\url{https://carlostoledo.co/reports/optconst-queue.html}.
\bibitem{robbins} H. Robbins, A remark on Stirling's formula,
\emph{Amer. Math. Monthly} \textbf{62} (1955), 26--29.
\end{thebibliography}
\end{document}