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M. Angeles Alfonseca

Publications and source records attributed to M. Angeles Alfonseca.

7 recordsLinked to original sources
arXiv2026

On Bezdek's conjecture for high-dimensional convex bodies with an aligned center of symmetry

In 1999, K. Bezdek posed a conjecture stating that among all convex bodies in $\mathbb R^3$, ellipsoids and bodies of revolution are characterized by the fact that all their planar sections have an axis of reflection. We prove Bezdek's conjecture in arbitrary dimension $n\geq 3$, assuming only that sections passing through a fixed point have an axis of reflection, provided that the complementary invariant subspaces are all parallel to a fixed hyperplane. The result is proven in both orthogonal and affine settings.

math.MG↗
arXiv2018

On bodies in $\mathbb{R}^5$ with directly congruent projections or sections

Let $K$ and $L$ be two convex bodies in ${\mathbb R^5}$ with countably many diameters, such that their projections onto all $4$ dimensional subspaces containing one fixed diameter are directly congruent. We show that if these projections have no rotational symmetries, and the projections of $K,L$ on certain 3 dimensional subspaces have no symmetries, then $K=\pm L$ up to a translation. We also prove the corresponding result for sections of star bodies.

math.MG↗
arXiv2015

On bodies with directly congruent projections and sections

Let $K$ and $L$ be two convex bodies in ${\mathbb R^4}$, such that their projections onto all $3$-dimensional subspaces are directly congruent. We prove that if the set of diameters of the bodies satisfy an additional condition and some projections do not have certain symmetries, then $K$ and $L$ coincide up to translation and an orthogonal transformation. We also show that an analogous statement holds for sections of star bodies, and prove the $n$-dimensional versions of these results.

math.MG↗
arXiv2015

Counterexamples Related to Rotations of Shadows of Convex Bodies

We construct examples of two convex bodies $K,L$ in $\mathbb{R}^n$, such that every projection of $K$ onto a $(n-1)$-dimensional subspace can be rotated to be contained in the corresponding projection of $L$, but $K$ itself cannot be rotated to be contained in $L$. We also find necessary conditions on $K$ and $L$ to ensure that $K$ can be rotated to be contained in $L$ if all the $(n-1)$-dimensional projections have this property.

math.MG↗
arXiv2013

On the local convexity of intersection bodies of revolution

One of the fundamental results in Convex Geometry is Busemann's theorem, which states that the intersection body of a symmetric convex body is convex. Thus, it is only natural to ask if there is a quantitative version of Busemann's theorem, i.e., if the intersection body operation actually improves convexity. In this paper we concentrate on the symmetric bodies of revolution to provide several results on the (strict) improvement of convexity under the intersection body operation. It is shown that the intersection body of a symmetric convex body of revolution has the same asymptotic behavior near the equator as the Euclidean ball. We apply this result to show that in sufficiently high dimension the double intersection body of a symmetric convex body of revolution is very close to an ellipsoid in the Banach-Mazur distance. We also prove results on the local convexity at the equator of intersection bodies in the class of star bodies of revolution.

math.MG↗